ENTRY GROUP THEORY Authors: started Mark Lezama: October 2003 Literature: "Algebra" by Michael Artin, Mathworld +------------------------------------------------------------ | Group theory +------------------------------------------------------------ Group theory is studies algebraic objects called groups. The German mathematician Karl Friedrich Gauss (1777-1855) developed but did not publish some of the mathematics of group theory. The French mathematician Evariste Galois (1811-1832) is generally credited with being the first to develop the theory, which he did by developing new techniques to study the solubility of equations. Group theory is a powerful method for analyzing abstract and physical systems in which symmetry --the intrinsic property of an object to remain invariant under certain classes of transformations-- is present because the mathematical study of symmetry is systematized and formalized in group theory. Consequently, group theory is an important tool in physics particularly in quantum mechanics. +------------------------------------------------------------ | group +------------------------------------------------------------ A group is an object consisting of a set G and a law of composition (or binary operation) L on G satisfying: L is associative. L has an identity in G. Every element of G has an inverse. The study of groups is known as group theory. If a group G has n elements where n is a positive integer, then G is a finite group with order n. If a group is not finite it is infinite. Examples: Z^+, the integers under addition; R^+ = (R, +), the real numbers under addition; R^x = (R - 0, .), the real numbers without zero under multiplication; GL_n(C), the nx n general linear group under matrix multiplication; S_n, the symmetric group on n objects under composition. +------------------------------------------------------------ | law of composition +------------------------------------------------------------ A law of composition or, binary operation, on a set S is a function from Sx S into S. That is, a law of composition on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, functional notation is not used; that is, if a law of composition f sends (a, b) to c, one does not usually write f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of real numbers, such as ab = c, a. b = c, a circ b = c, a + b = c, and so on. An example of a law of composition is multiplication on the real numbers, R. If m colon R x R to R defines multiplication on R then m(x, y) = x . y. For example m(2, 5) = 2 . 5 = 10. +------------------------------------------------------------ | binary operation +------------------------------------------------------------ A binary operation, or law of composition, on a set S is a function from S x S into S. That is, a binary operation on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, functional notation is not used; that is, if a binary operation f sends (a, b) to c, one does not usually write f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of real numbers, such as ab = c, a. b = c, a circ b = c, a + b = c, and so on. An example of a binary operation is multiplication on the real numbers, R. If mcolon R x R to R defines multiplication on R then m(x, y) = x . y. For example m(2, 5) = 2 . 5 = 10. +------------------------------------------------------------ | associative +------------------------------------------------------------ A law of composition on a set S is associative if for all a, b, c in S, (ab)c = a(bc). The informal intuition behind associativity (the property of being associative) is that if one has an expression in which there are many parentheses and the only operation performed in this expression is that defined by an associative law of composition, then one may ignore the parentheses. For example, if . is an associative law of composition on S and a, b, c, d in S, then ((a.(b. c)). d = ((a. b). c) . d = (a. b) . (c . d) and so on; thus one may write a . b . c . d without being ambiguous. An example of an associative law of composition is addition on the integers, Z. That is, for all a, b, c in Z, (a + b) + c = a + (b + c). +------------------------------------------------------------ | identity +------------------------------------------------------------ An identity for a law of composition on a set S is an element e such that, for all ain S, ea = a and ae = a. Note that a law of composition has at most one identity. The symbols e, 0 and 1 are commonly used to denote the identity element of a group. The number 0 is an identity for addition on the real numbers. +------------------------------------------------------------ | identity +------------------------------------------------------------ Suppose a set S has a law of composition with identity 1. For every element ain S, if there exists an element bin S such that ab = 1 and ba = 1 then b is the inverse of a. When using multiplicative notation for the law of composition, the inverse of a can be written as a^-1. As an example, the inverse of any integer n is -n where the law of composition is addition and the identity is 0. As another example, the inverse of any nonzero real number x is frac1 x, where the law of composition is multiplication and the identity is 1. +------------------------------------------------------------ | general linear group +------------------------------------------------------------ The n x n general linear group GL_n(F) is the set of n x n matrices with entries in the field F and nonzero determinant, under the law of composition of matrix multiplication. Thus GL_n(F) is the group of n x n invertible matrices with entries in F. If F is a finite field of field order q then sometimes the general linear group GL_n(F) is denoted by GL_n(q). The general linear group often appears with respect to the real numbers, R, or the complex numbers, C; that is, the general linear group often appears as GL_n(R) or GL_n(C). The special linear group SL_n(F) is the subgroup of GL_n(F) whose elements have determinant equal to 1. +------------------------------------------------------------ | special linear group +------------------------------------------------------------ The n x n special linear group SL_n(F) is the set of n x n matrices with entries in the field F and determinant equal to 1, under the law of composition of matrix multiplication. If F is a finite field of field order q then sometimes the special linear group SL_n(F) is denoted by SL_n(q). SL_n(F) is a subgroup of the general linear group GL_n(F). +------------------------------------------------------------ | trivial +------------------------------------------------------------ A group is trivial if it contains exactly one element. The one element in the group is the identity element. As all trivial groups are isomorphic, one usually refers to a trivial group as emphthe trivial group. A group that is not trivial is nontrivial. +------------------------------------------------------------ | trivial +------------------------------------------------------------ The group containing exactly one element (the identity) is unique up to isomorphism and is therefore called the trivial group. The trivial group is a normal subgroup of every group. +------------------------------------------------------------ | nontrivial +------------------------------------------------------------ A group is nontrivial if it is not trivial. +------------------------------------------------------------ | abelian +------------------------------------------------------------ A group is abelian if its law of composition is commutative. Examples of abelian groups include the following: R^+ = (R, +), the real numbers under addition; R^x = (R - 0, .), the real numbers without zero under multiplication; any cyclic group. Examples of nonabelian groups, i.e. groups that are not abelian: GL_n(C), the general linear group; The symmetric group on n objects, where n is a positive integer greater than 2. +------------------------------------------------------------ | commutative +------------------------------------------------------------ A law of composition on a set S is commutative if for all a, b in S ab = ba. An example of a commutative law of composition is addition on the real numbers: for example, 3.2 + 4 = 7.2 = 4 + 3.2. +------------------------------------------------------------ | cancellation Law +------------------------------------------------------------ The cancellation Law states that if a, b, and c are elements of a group and if ab = ac then b = c. Similarly, if ba = ca then b = c. The Cancellation Law follows from the fact that every element of a group has an inverse. +------------------------------------------------------------ | permutation +------------------------------------------------------------ If S is a set, then a permutation of S is a bijective map from S into S. The intuition underlying the definition of a permutation is that a permutation determines a reordering of the elements in a list or the rearrangement of objects. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 1, and sigma(3) = 3 can be thought to represent the reordering of the list 1,2,3 that results in the list 2,1,3. There is an important kind of permutation called a transposition. A transposition of a set S is a permutation sigmacolon S to S satisfying the following: there exist s_1, s_2 in S such that sigma(s_1) = s_2 sigma(s_2) = s_1 and for all sin S, if s neq s_1 and s neq s_2, then sigma(s) = s. Every permutation of a finite set can be written as the composition of a finite number of transpositions of that set. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 3, and sigma(3) = 1 is equivalent to the composition of two transpositions. Define sigma_1: 1, 2, 3 to 1, 2, 3 by sigma_1(1) = 2, sigma_1(2) = 1, and sigma(3) = 3, and define sigma_2: 1, 2, 3 to 1, 2, 3 by sigma_2(1) = 3, sigma_1(2) = 2, and sigma(3) = 1. Then sigma_1 and sigma_2 are transpositions and sigma = sigma_2 circ sigma_1. The sign of a permutation sigma is (-1)^n where n is a finite positive integer such that there exist n transpositions whose composition equals sigma. If a permutation has sign 1, then it is called an even permutation. If a permutation has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition). The sign map encapsulates the notion of the sign of a permutation of a finite set. The set of permutations on a set forms a group where the law of composition is composition of functions. One example of a group of permutations that appears frequently in group theory is the symmetric group on n objects, i.e. the group of permutations of the set 1, 2, ldots, n. +------------------------------------------------------------ | transposition +------------------------------------------------------------ A transposition of a set S is a permutation sigmacolon S to S satisfying the following: there exist s_1, s_2 in S such that sigma(s_1) = s_2 sigma(s_2) = s_1 and for all sin S, if s neq s_1 and s neq s_2, then sigma(s) = s. Every permutation of a finite set can be written as the composition of a finite number of transpositions of that set. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 3, and sigma(3) = 1 is equivalent to the composition of two transpositions. Define sigma_1: 1, 2, 3 to 1, 2, 3 by sigma_1(1) = 2, sigma_1(2) = 1, and sigma(3) = 3, and define sigma_2: 1, 2, 3 to 1, 2, 3 by sigma_2(1) = 3, sigma_1(2) = 2, and sigma(3) = 1. Then sigma_1 and sigma_2 are transpositions and sigma = sigma_2 circ sigma_1. Every transposition is an odd permutation. +------------------------------------------------------------ | sign +------------------------------------------------------------ The sign of a permutation sigma is (-1)^n where n is a finite positive integer such that there exist n transpositions whose composition equals sigma. If a permutation has sign 1, then it is called an even permutation. If a permutation has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition). +------------------------------------------------------------ | even permutation +------------------------------------------------------------ An even permutation is a permutation that has sign 1. That is, an even permutation is the composition of an even number of transpositions. Thus the identity permutation is an even permutation. +------------------------------------------------------------ | odd permutation +------------------------------------------------------------ An odd permutation is a permutation that has sign -1. That is, an odd permutation is the composition of an odd number of transpositions. Thus every tranposition is an odd permutation. +------------------------------------------------------------ | symmetric group +------------------------------------------------------------ The symmetric group on n objects, denoted S_n, is the group of permutations of the set 1, 2, ldots, n; the law of composition is composition of functions. The order of S_n is n! for all positive integers n. For example, S_2 = e, sigma, where e is the identity permutation, and sigma is a transposition. That is, e is the identity element of S_2 and is defined by e(1) = 1 and e(2) = 2; sigma is defined by sigma(1) = 2 and sigma(2) = 1. +------------------------------------------------------------ | sign map +------------------------------------------------------------ The sign map, denoted sign, is a group homomorphism from the symmetric group, S_n, into the group 1, -1 (under multiplication). The sign map is defined by sign(sigma) = (-1)^k where sigma is equal to the composition of k transpositions. The sign map is well-defined because it is a standard result that if sigma is any permutation (of any set), and if sigma is equal to the composition of k transpositions and is also equal to the composition of m transpositions, then (-1)^k = (-1)^m. The kernel of the sign map is the alternating group, A_n; that is, A_n is the group of even permutations on n objects. +------------------------------------------------------------ | alternating group +------------------------------------------------------------ The alternating group is the kernel of the sign map. In other words, the alternating group on n objects, usually denoted A_n, is a normal subgroup of the symmetric group on n objects: A_n = sigma in S_n mid sign(sigma) = 1 . Thus A_n is the group of even permutations in S_n. For n geq 5, A_n is a simple group, i.e. a group that has no proper normal subgroup. +------------------------------------------------------------ | simple group +------------------------------------------------------------ A group G is a simple group if every normal subgroup N of G is not a proper subgroup. That is, G is simple if its only normal subgroups are G and the trivial group. The alternating group A_n is simple for n geq 5. Any cyclic group of prime order is simple. Any cyclic group of prime order is simple. In fact, any simple abelian group is a cyclic group of prime order. +------------------------------------------------------------ | subgroup +------------------------------------------------------------ A subset H of a group G is a subgroup of G if it satisfies the following properties: If ain H and bin H, then ab in H. 1in H, where 1 is the identity element of G. If ain H then the inverse of a, a^-1, is also in H. When it is clear that G is a group, sometimes H subseteq G is used to denote that H is a subgroup of G (as opposed to merely being a subset of G). Every nontrivial group G has at least two subgroups: the whole group G and the subgroup 1 consisting exactly of the identity element of G. If G is trivial then these two subgroups are the same and G has exactly one subgroup. A subgroup is a proper subgroup if it is neither the whole group nor the trivial group. As an example, the integers under addition are a subgroup of the real numbers under addition. By Lagrange's Group Theorem, if H is a subgroup of a finite group G, the order of H divides the order of G. +------------------------------------------------------------ | proper subgroup +------------------------------------------------------------ A proper subgroup of a group G is a nontrivial subgroup of G that is not equal to G. +------------------------------------------------------------ | order +------------------------------------------------------------ A finite group G is said to have order n if G has n elements. More generally, the order of a group G is the cardinality of the set G, both of which are often denoted |G|. For any given element x of a given group, if there exists a positive integer k such that x^k = 1, then x is said to have order m, where m is the least positive integer satisfying x^m = 1. If x^k neq 1 for all positive integers k, then x is said to have infinite order. +------------------------------------------------------------ | cyclic group +------------------------------------------------------------ If G is a group and if x is an element of G, the cyclic group langle x rangle generated by x is the set of all powers of x: langle x rangle = ldots, x^-2, x^-1, 1, x, x^2, ldots. Note that langle x rangle is the smallest subgroup of G which contains x. Further note that any cyclic group is abelian. If x has infinite order then langle x rangle is said to be infinite cyclic. Note that if langle x rangle is infinite cyclic then langle x rangle is isomorphic to Z^+, the integers under addition. As a result, one sometimes refers to any infinite cyclic group as emphthe infinite cyclic group, denoted Z^+. If x has order n, then langle x rangle has order n and is called a cyclic group of order n: langle x rangle = 1, x, x^2, ldots, x^n. If langle x rangle is a cyclic group of order n, then langle x rangle is isomorphic to Z/n, where Z/n is the group satisfying the following properties (we use additive notation as opposed to multiplicative notation for the law of composition of Z/n): 1. Z/n = 0, 1, 2, ldots, n - 1. 2. for any x, y in Z/n, x +_1 y is the unique element in Z/n which is congruent modulo n to x + y, where +_1 denotes the law of composition on Z/n and + denotes conventional integer addition. Normally + is used to denote the law of composition on Z/n, but +_1 is used here to distinguish it from conventional addition. Two integers a and b are congruent modulo n, written aequiv b (mboxmodulo n), if n divides b - a. As a result, one sometimes refers to any cyclic group of order n as emphthe cyclic group of order n, often denoted Z/n. +------------------------------------------------------------ | cyclic group +------------------------------------------------------------ Let G_1 and G_2 be groups. A map varphicolon G_1 to G_2 is a group homomorphism if varphi(ab) = varphi(a)varphi(b) for all a,bin G_1. Here we use the same multiplicative notation for the laws of composition of G_1 and G_2, even though there is no requirement that their laws of composition be the same. Note that varphi(1_G_1)= 1_G_2 and varphi(a^-1) = varphi(a)^-1 for all ain G, where 1_G_i is the identity of G_i. The kernel of varphi, sometimes denoted ker varphi, is the set xin G_1 mid varphi(x) = 1_G_2 . Note that ker varphi is a normal subgroup of G_2. Another subgroup of G_2 determined by varphi is the image of varphi, sometimes denoted im varphi. The image of varphi is im varphi = xin G_2 mid x = varphi(a) for some a in G_1. Sometimes the image of varphi is denoted varphi(G_1). The following are examples of homomorphisms: The inclusion map icolon H to G defined by i(a) = a, where H is a subgroup of G. ker i = 1_G and im i = H. For a fixed a in G, the map varphicolon Z^+ to G defined by varphi(n) = a^n, where Z^+ denotes the integers with addition. ker varphi = n mid a^n = 1_G and im varphi = langle a rangle (the cyclic subgroup generated by a). The determinant map detcolon GL_n(R) to R^x, where GL_n(R) denotes the general linear group and R^x denotes the real numbers without zero under multiplication. ker det = SL_n(R), the special linear group, and im det = R^x. The sign map on permutations signcolon S_n to 1, -1, where S_n denotes the symmetric group on n objects. ker sign = A_n, the alternating group, and im sign = 1, -1. Let R_1 and R_2 be rings. A map varphicolon R_1 to R_2 is a ring homomorphism if varphi(a + b) = varphi(a) + varphi(b), varphi(ab) = varphi(a)varphi(b), and varphi(1_R_1) = 1_R_2, for all a,bin R_1. Here we use the same additive and multiplicative notation for the laws of composition of R_1 and R_2, even though there is no requirement that their laws of composition be the same. +------------------------------------------------------------ | kernel +------------------------------------------------------------ The kernel of a group homomorphism varphicolon G_1 to G_2 is the set ker varphi = x in G_1 mid varphi(x) = 1_G_2 , where 1_G_2 denotes the identity of G_2. The kernel of a homomorphism is an important example of a normal subgroup. There are many results involving the kernel of a homomorphism. +------------------------------------------------------------ | image +------------------------------------------------------------ The image of a map varphicolon G_1 to G_2 is the set xin G_2 mid x = varphi(a) for some a in G_1. In general, the image of varphi is often denoted varphi(G_1). If varphi is a group homomorphism, then the image of varphi is a subgroup of G_2 and is sometimes denoted im varphi. +------------------------------------------------------------ | isomorphism +------------------------------------------------------------ A group isomorphism is a bijective group homomorphism. A ring isomorphism is a bijective ring homomorphism. +------------------------------------------------------------ | isomorphic +------------------------------------------------------------ Two groups G_1 and G_2 are isomorphic if there exists a group isomorphism from G_1 into G_2. Sometimes G_1 cong G_2 is used to denote `G_1 and G_2 are isomorphic.' Note that cong is an equivalence relation on the set of all groups. When one speaks of classifying groups, hat is usually referred to is the classification of isomorphism classes. Thus one might say that there are two groups of order 6 emphup to isomorphism, meaning that there are two isomorphism classes of groups of order 6. One sometimes says that `G_1 is isomorphic to G_2' instead of saying `G_1 and G_2 are isomorphic.' +------------------------------------------------------------ | automorphism +------------------------------------------------------------ An automorphism of a group G is an isomorphism from G into G. The identity map is a simple example of an automorphism. Conjugation by an element of the group is an important example of an automorphism. That is, for a fixed element bin G, conjugation by b is the map varphicolon Gto G defined by varphi(a) = bab^-1. Here we use multiplicative notation for the group law of composition. Note that if G is abelian, then conjugation by any element is the identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a conjugation not equal to the identity map) of G. +------------------------------------------------------------ | automorphism +------------------------------------------------------------ Let G be a group and let bin G. The map varphicolon Gto G defined by varphi(a) = bab^-1 is conjugation by b. Note that conjugation is an automorphism of G. Further note that if G is abelian, then conjugation by any element is the identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a conjugation not equal to the identity map) of G. +------------------------------------------------------------ | conjugation +------------------------------------------------------------ Let G be a group and let H be a subgroup of G. H is a normal subgroup of G (sometimes written H triangleleft G) if for all a in H and for all x in G, xax^-1in H. Note that it follows that any subgroup of an abelian group is normal. Normal subgroups appear often in group theory. Every group G has at least one normal subgroup, called the center of G, denoted by Z or Z(G). The center of G is the set of elements that commute with every element of G: Z(G) = zin G mid zx = xz for all xin G. Another important example of a normal subgroup is the kernel of a group homomorphism. +------------------------------------------------------------ | center +------------------------------------------------------------ The center of a group G, denoted by Z or Z(G) is the set of elements that commute with every element of G: Z(G) = zin G mid zx = xz for all xin G. Note that if G is abelian then Z(G) = G. +------------------------------------------------------------ | coset +------------------------------------------------------------ Given a subgroup H of a group G, a coset of H is a subset H' of G such that there exists an a in G such that (1) H' = aH = ah mid h in H, in which case H' is said to be a left coset; or (2) H' = Ha = ha mid h in H, in which case H' is said to be a right coset. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G: kervarphi = |im varphi|. Thus another result of Langrange's Theorem is that |G| = |ker varphi| . |im varphi|. +------------------------------------------------------------ | left coset +------------------------------------------------------------ Given a subgroup H of a group G, a left coset of H is a subset H' of G such that there exists an a in G such that H' = aH = ah mid h in H. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G: ker(varphi) = |im(varphi)|. Thus another result of Langrange's Theorem is that |G| = |ker(varphi)| . |im(varphi)|. +------------------------------------------------------------ | right coset +------------------------------------------------------------ Given a subgroup H of a group G, a right coset of H is a subset H' of G such that there exists an a in G such that H' = Ha = ha mid h in H. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G:ke(varphi) = |im(varphi)|. Thus another result of Langrange's Theorem is that |G| = |ker(varphi)| . |im(varphi)|. +------------------------------------------------------------ | index +------------------------------------------------------------ The index of subgroup H of a group G is the cardinality of the set of left cosets of H in G. The index of H in G is denoted G:H. +------------------------------------------------------------ | quotient group +------------------------------------------------------------ Given a group G and a normal subgroup N of G, the quotient group of N in G, written G / N and read ``G mod(ulo) N'', is the set of cosets of N in G, under the law of composition that is defined as follows: (aN)(bN) = abN, where xN = xn mid n in N. Note that since N is normal, aN = Na for all a in G, so it is not necessary to define this law of composition in terms of left cosets instead of right cosets. The order of G / N is the index G:N of N in G. Quotient groups can be identified by the First Isomorphism Theorem: if varphi colon G to G' is a surjective group homomorphism and if N = ke(varphi) then psicolon G / N to G' is an isomorphism, where psi is defined by psi(aN) = varphi(a). +------------------------------------------------------------ | First Isomorphism Theorem +------------------------------------------------------------ The First Isomorphism Theorem. Suppose varphi colon G to G' is a surjective group homomorphism, and let N denote the kernel of varphi. Then the quotient group G / N is isomorphic to G' by the map psi defined by psi(aN) = varphi(a). The First Isomorphism Theorem is the principle method of identifying quotient groups. As an example, consider the group homomorphism varphi from C^x, the nonzero complex numbers under multiplication, into R^x, the nonzero real numbers under multiplication, defined by varphi(z) = |z|, where |z| denotes the absolute value of z. The kernel of varphi is the unit circle, U, and the image of varphi is the group of positive real numbers. So C^x / U is isomorphic to the multiplicative group of positive real numbers. +------------------------------------------------------------ | operation +------------------------------------------------------------ Given a group G and a set S, an operation of a G on S is a map from G x S into S - often written using multiplicative notation: (g,s) mapsto gs - satisfying: 1 s = s for all s in S, where 1 is the identity of G; and (g g')s = g(g's), for all g, g' in G and for all s in S. There are some terms that are sometimes associated with a group operation: S is often called a G-set; G is sometimes called a transformation group; and the group operation is often also called a group action. Mathworld: "Historically, the first group action studied was the action of the Galois group on the roots of a polynomial. However, there are numerous examples and applications of group actions in many branches of mathematics, including algebra, topology, geometry, number theory, and analysis, as well as the sciences, including chemistry and physics." This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 44 entries in this file. COUNT: 44