ABSTRACT
This section starts by considering group homomorphisms, by way of analogy with ring homomorphisms: A function between groups ϕ : G→ S is a group homomorphism if ϕ(gh) = ϕ(g)ϕ(h) for every g, h ∈ G. A group homomorphism preserves the group identity and inverses, the image of G is a subgroup of S, and if G is abelian then so is ϕ(G) (Theorem 28.1). If ϕ is one-to-one and onto, then we say ϕ is an isomorphism, in which case ker(ϕ) = 1G. A subgroup H of G is normal if gH = Hg for all g ∈ G; that is, if
the left and right cosets of H are the same for each element of G. This is equivalent to saying that g−1Hg ⊆ H for all g ∈ G. If H is normal in G, then the collection of cosets of H in G, denoted G/H, forms a group under the operation (Ha)(Hb) = Hab (Theorem 32.1). G/H is then called the group of cosets of G mod H or the quotient group of G mod H. Normal subgroups are to groups what ideals are to rings. The kernel ker(ϕ) of any group homomorphism is a normal subgroup of G. Paralleling rings, there is the Fundamental Isomorphism Theorem
for Groups (Theorem 33.4) which says that if ϕ : G → S is an onto homomorphism, then G/ ker(ϕ) is isomorphic to S. Whether the subgroup H is normal in G or not, the number of left
cosets of H is the same as the number of right cosets. Assuming G is finite, this number is called the index of H in G and denoted by [G : H]. Lagrange’s Theorem (Theorem 31.2) says that |G| = [G : H]|H|. Thus the order of a subgroup must divide the order of the group and so the order of any element must divide the order of the group. Thus, any group of prime order is cyclic. An important group is the group of permutations on the set {1, 2, . . . , n}, also called the nth symmetric group and denoted by Sn. Clearly |Sn| = n!. Cayley’s theorem (Theorem 29.1) says that every group of order n is isomorphic to a subgroup of Sn. A transposition is a cycle of length two. Any permutation can be
factored as a product of transpositions (Theorem 34.1), and, while this factorization is not unique, all factorizations of a given permutation a permutation as
n called the alternating group. The alternating group An is a normal subgroup of the symmetric group Sn and [Sn : An] = 2. Furthermore, An is simple for n 6= 4. All finite abelian groups can be completely described by the Fun-
damental Theorem for Finite Abelian Groups (Theorem 35.1): Every finite abelian group is isomorphic to a direct product of cyclic groups; each cyclic group in this decomposition is of order pn, where p is prime. That is, each finite abelian group is isomorphic to a group of the form
× · · · × Z pknn
where the pi’s are primes (not necessarily distinct), and the ki’s are positive integers (not necessarily distinct). It follows from this theorem that if G is a finite abelian group of
order r and m divides r, then G has a subgroup of order m (Corollary 35.2). That is, G has subgroups of every possible order. A p-group is a group where each of its elements has order a power of
the prime p. If G is a finite abelian p-group, then |G| = pn, for some n (Corollary 35.3). Finally, the idea of abelian group is generalized by the idea of solvable
groups. A group is solvable if it has a finite collection of subgroups G0, G1, · · ·Gn, so that
Gn = {1} ⊆ Gn−1 ⊆ Gn−2 ⊆ · · · ⊆ G1 ⊆ G0 = G,
and furthermore each Gi+1 is normal in Gi, and the group of cosets Gi/Gi+1 is abelian. Such a sequence of subgroups is called a subnormal series with abelian quotients for G. An important group that is not solvable is S5 (Example 36.4). This important fact we will use in Section IX. Every homomorphic image of a solvable group is also solvable (The-
orem 36.1) as is every subgroup of a solvable group (Theorem 36.2). In fact, if G has a normal subgroup H, then G is solvable if and only if both H and G/H are solvable (Theorem 36.3).
