By Randall R. Holmes

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By the previous corollary, |G| = o(x)n for some n ∈ N. Thus x|G| = (xo(x) )n = en = e. 5 – Exercises 5–1 Let G be a group and let H and K be subgroups of G. For x ∈ G, put HxK = {hxk | h ∈ H, k ∈ K} (called the (H, K) double coset of G determined by x). (a) Define a relation on G by putting x ∼ y if x = hyk for some h ∈ H and k ∈ K. Prove that ∼ is an equivalence relation and that the equivalence class of x ∈ G relative to this equivalence relation is precisely HxK. ) 33 (b) Prove that for each x ∈ G, the double coset HxK is a union of right cosets of H and the cardinality of the set of these cosets is |K : K ∩H x |, where H x = x−1 Hx = {x−1 hx | h ∈ H}.

1 Definition Let G be a group. A subnormal series of G is a tuple (Gi ) = (G0 , G1 , . . Gr ), where • G0 = G, • Gr = {e}, • Gi+1 Gi for all 0 ≤ i < r. Thus, a subnormal series of G is a finite descending sequence of subgroups of G starting at G and ending at {e} with each term normal in the preceding term. Let (Gi ) = (G0 , G1 , . . , Gr ) be a subnormal series of G. The factors of (Gi ) are the quotient groups Gi /Gi+1 , 0 ≤ i < r (or any list of r groups isomorphic to these quotient groups).

If aN, bN ∈ G/N , then (aN )(bN ) = abN b N = abN N = abN using normality of N . This gives part (i). Associativity of coset multiplication now follows immediately from the associativity of multiplication in G. Also, that eN is an identity and that a−1 N is an inverse of aN ∈ G/N are immediate, so (ii) follows. (G/N, · ) is the quotient group (or factor group) of G by N . The notation G/N is read “G modulo N ” or “G mod N ”. The normality assumption on N is essential here. In fact, if H ≤ G is not normal, then the set of left cosets of H in G is never closed under set multiplication (see Exercise 7–1).

### Algebra I by Randall R. Holmes

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