Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping

  • Thread starter Poopsilon
  • Start date
  • #1
294
1

Homework Statement



Show that there is no non-abelian group [itex]G[/itex] such that [itex]Z(G)=\mathbb{Z}_2[/itex], which satisfies the short exact [itex]\mathbb{Z}_2\rightarrow G\rightarrow\mathbb{Z}_2^3[/itex].


The Attempt at a Solution



I have knowledge of group theory up through proofs of the Sylow theorems. I know the center is contained in every normal subgroup of G. [itex]\mathbb{Z}_2^3[/itex] has a seven subgroups of order 2 so I've been trying to use the correspondence theorem to get some idea of what this implies for the structure of G, but no luck so far. I've found several paths to the fact that G has no element of order 8, but that still leaves a lot of possibilities for its subgroup of order 8. Anyways I've been banging my head against this one for a while now, can anyone help me out with it? Thanks.

Note: I want to prove this without resorting to the classification of groups of order 16.
 
Last edited:

Answers and Replies

Related Threads on Show no non-abelian group G such that Z(G)=Z2 exists satisfying the mapping

  • Last Post
Replies
2
Views
1K
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
14
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
554
  • Last Post
2
Replies
38
Views
3K
Top