Proofs : Grorups, Subgroups and Indexes

1.) If |G|=p^n for some prime p, and 1 is not equal to H, a normal subgroup of G, show that H intersect Z(G) is not equal to 1. 2.) Suppose G is finite, H is a subgroup of G, [G:H]=n and |G| does not divide n!. Show that there is a normal subgroup K of G, with K not equal to 1, such that K is a subgr ...continues

Subgroup proofs

I need to see rigorous proofs of these propositions. 1.) Suppose [G:H] is finite. Show that there is a normal subgroup K of G with K, a subgroup of H, such that [G:K] is finite. 2.) Suppose H is a subgroup of S_n but H is not a subgroup of A_n. Show that [H:A_n intersect H]=2. 3.) Prove that if H,K are ...continues

Subgroup proof

Suppose that H and K both have finite index in G. Prove that [G:H n K]<=[G:H][G:K].

Normal subgroup proof

Let |G| be finite and N be a normal subgroup of G. If xN is an element of G/N and has order a power of p, show that there exists a y element of G such that |y| is a power of p and yN=xN.

Prove or Give a counterexample:

1.) G_1 is isomorphic to G_2 and H_1 is isomorphic to H_2 implies G_1/H_1 is isomorphic to G_2/H_2 2.) G_1 is isomorphic to G_2 and G_1/H_1 is isomorphic to G_2/H_2 implies H_1 is isomorphic to H_2 3.) H_1 is isomorphic to H_2 and G_1/H_1 is isomorphic to G_2/H_2 implies G_1 is isomorphic to G_2

Groups

Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic to K_1xK_2 and no H_i is isomorphic to any K_j. Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of any two of its proper subgroups. If G is the internal direct product of subgroups G_1 and G_2 show that G/G_ ...continues

Solvable groups

1.) Show that Z(II_alpha G_alpha)=II_alpha Z(G_alpha) 2.) Show that (G_1 x G_2 x ... x G_n)' = G'_1 x G'_2 x ... x G'_n 3.) Under what circumstances is G_1 x G_2 x ... x G_n solvable? Note that II_alpha is the product over all alpha of an index set.

Unique p-Sylow subgroup

If G/N is abelian and P is a p-Sylow subgroup of G, prove that PN/N is the unique p-Sylow subgroup of G/N.

free products and pushouts

please attached file for question Suppose that....are pushouts...

Short exact sequences

A pair of homomorphisms is said to be exact...(see attached)