Let G be a group and let H, K be subgroups of G such that hk=kh for all h in H a

Question

Let G be a group and let H, K be subgroups of G such that hk=kh for all h in H and k in H, and that H intersect K = the identity, and HK = G.

a) Prove: every element in G can be written as a unique product hk, h in H and k in K.

b) In this case, we sometimes call G an internal direct product of H and K. Prove that G is isomorphic to the (external) direct product H x K .

c) We know that Zn x Zm is isomorphic to Znm if gcd (n,m)= 1. How would Znm be realized as an internal direct product of subgroups isomorphic to Zn and Zm? (Hint do some particular cases, like Z6 and Z15, and show these examples)

d)Is it possible that G is an internal direct product of two non-abelian groups? An example would suffice.

Explanation / Answer

... H, k ? K }. (a) Show that HK is a subgroup of G if and only if HK = KH. ... (c) Give an example of a group G and two subgroups H and K such that HK is not a subgroup ... Also for the same elements h ? H and k ? K, we have (hk)?1 ? HK, so. (hk)?1 .... Conversely suppose Hx = xH for all x ? G. Let h ? H and x ? G. Then ...

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