do these two groups isomorphic to each other?

Question

consider two groups $T=<x,y |x^4=y^3=1,yxy=x>$ and $A=<x,y |x^6=1,x^3=y^2,xy=yx^{-1}>$, are these two groups isomorphic?

I think this is not true,because $T$ don't have any 4 member cyclic subgroups but $A$ has $\mathbb{Z}_{4}$ as a cyclic subgroups.

so if they are not isomorphism then can you tell me what group $A$ is isomorphic to within 12 elements nonabelian groups? thank you very much.

score 1 · Accepted Answer · answered Jun 14 '15 at 15:44

1

$T$ is the semidirect product of $C_3$ by $C_4$ by the map $g : C_4 \rightarrow Aut(C_3)$ given by $g(k) = a^k$, where $a$ is the automorphism $a(x) = -x$. It is indeed isomorphic to $A$. So, $A\simeq T$ are isomorphic nonabelian groups of order $12$. Altogether there are three different nonabelian groups of order $12$, namely $D_6$, $A_4$ and $T$.

answered Jun 14 '15 at 15:44

Dietrich Burde

140,055

why checking their subgroups as I did, give wrong result? – kpax Jun 14 '15 at 15:48
my mistake!!both have it! – kpax Jun 14 '15 at 15:49
yes,you are right, thanks for your good answer! – kpax Jun 14 '15 at 15:51

do these two groups isomorphic to each other?

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