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H.Shahsavari
H.Shahsavari
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Let $G$ be a finite group with the following property:

For every nontrivial proper subgroups $H$ and $K$ infor which $H\cap K=1$, if the number of their minimal subgroups are $m$ and $n$ respectively, then the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$, where $H\cap K=1$. What

What are the possible structures for $G$? I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?

Let $G$ be a finite group with the following property:

For every nontrivial proper subgroups $H$ and $K$ in which the number of their minimal subgroups are $m$ and $n$ respectively, the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$, where $H\cap K=1$. What are the possible structures for $G$? I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?

Let $G$ be a finite group with the following property:

For every nontrivial proper subgroups $H$ and $K$ for which $H\cap K=1$, if the number of their minimal subgroups are $m$ and $n$ respectively, then the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$.

What are the possible structures for $G$? I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?

Source Link
H.Shahsavari
H.Shahsavari
  • 479
  • 2
  • 12

Minimal Subgroups

Let $G$ be a finite group with the following property:

For every nontrivial proper subgroups $H$ and $K$ in which the number of their minimal subgroups are $m$ and $n$ respectively, the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$, where $H\cap K=1$. What are the possible structures for $G$? I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?