2 votes
Accepted

In the decomposition of $G/Z(G)$ into direct product of cyclic groups at least two factors of maximal orders must accur?

Let $G$ be a (non-abelian) $p$-group with $G/Z(G)$ abelian. Then $G' \subseteq Z(G)$, so certainly $G$ is nilpotent of class $2$. Put $p^a=$exp$(G/Z(G))$ (which equals the maximum order of the ...
Nicky Hekster's user avatar
2 votes

Suppose $H$ is a subgroup of $G$ and $H \approx \overline{H}$. It is "correct" to say $\overline{H}$ is a subgroup of $G$?

Suppose $H$ is a subgroup of $G$ and $H \approx \overline{H}$. It is "correct" to say $\overline{H}$ is a subgroup of $G$? Not strictly, no. This is evident in any infinite group $G$ that ...
Shaun's user avatar
  • 45.2k
1 vote

Suppose $H$ is a subgroup of $G$ and $H \approx \overline{H}$. It is "correct" to say $\overline{H}$ is a subgroup of $G$?

The question is not asking you about an isomorphism, but about actual subgroups of $S_4 \oplus S_4$ that don't have a direct sum decomposition inherited from the direct sum decomposition of $S_4 \...
KCd's user avatar
  • 46.2k

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