The Abelianization of a Group: Implications and Proofs

[Newtime]

I know the definition of the Abelianization and the most direct implications of this new group, but what I don't understand are some of the broader implications. For instance, what can we say about the number of generators of the abelianization of a group with respect to the number of generators of that group? Also, one quick question: is it true that the abelianization of the direct product of groups is the direct product of the abelianization of the components (the direct product is finite)?

 

[quasar987]

If a group G has S as a generating set, then for any normal subgroup N of G, SN={sN:s in S} is a generating set for G/N, and of course, |SN|<=|S|. If S is finite, then equality holds iff for all s, s' in S, $s^{-1}s'\notin N$.

This holds for any group. Were you looking for a sharper result specifically in the case where N=[G,G]=G' (the derived subgroup)?

As for your second question, it is not difficult and you should try answering it yourself by splitting the problem into simpler ones:
A) Show (GxH)/(NxM) ~ (G/N) x (H/M) for any group G,H and normal subgroups N,M (where ~ means isomorphic)
B) Show (GxH)'=G'xH' for any groups G,H
C) Conclude from this.

 

[Newtime]

If a group G has S as a generating set, then for any normal subgroup N of G, SN={sN:s in S} is a generating set for G/N, and of course, |SN|<=|S|. If S is finite, then equality holds iff for all s, s' in S, $s^{-1}s'\notin N$.

This holds for any group. Were you looking for a sharper result specifically in the case where N=[G,G]=G' (the derived subgroup)?

As for your second question, it is not difficult and you should try answering it yourself by splitting the problem into simpler ones:
A) Show (GxH)/(NxM) ~ (G/N) x (H/M) for any group G,H and normal subgroups N,M (where ~ means isomorphic)
B) Show (GxH)'=G'xH' for any groups G,H
C) Conclude from this.

Thanks for the help; you've given me some good stuff to work out, and it looks like the result I was looking for is not only true, but shouldn't be too hard to prove.