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I don't know whether it is a good title for this question or not. Please if possible suggest a different title.

Let $G$ be a finite group and $L$ be a minimal subgroup of $G$ of order $p$ for some prime $p$. If there is only one nontrivial proper subgroup of $G$ containing properly $L$, then one of the following cases occurs:

a) $G\cong D_{8}$

b) $G\cong Z_{p}\times Z_{q^2}$, for some primes(not necessarily distinct) $p, q$

c) $G\cong Z_{p}\ltimes(Z_{p}\times Z_{p})$, for some prime $p\neq2$, where $Z_{p}$ acts nontrivially on $Z_{p}\times Z_{p}$

d) $G\cong Z_{q}\ltimes(Z_{p}\times Z_{p})$, for some primes $p, q$ and $q\mid(p^{2}-1)$, where $Z_{q}$ acts nontrivially on $Z_{p}\times Z_{p}$

e) $G\cong L_{2}(q)$, for some prime $q$ and $q\not\equiv \pm 1, \pmod 8$.

Some examples of the last case, (e) is $L=Z_{5}$ in $G=A_{5}$, $L=Z_{11}$ in $G=L_{2}(11)$, $L=Z_{7}$ in $G=L_{2}(13)$ and etc.

Is there any other possible structure for $G$?

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  • $\begingroup$ This question has a strange form. You make a statement, and then ask if it is true. $\endgroup$
    – Geoff Robinson
    Commented Jan 26, 2017 at 9:18
  • $\begingroup$ What about $L=Z_{13}$ inside $PSL_3(3)$? $\endgroup$
    – Nick Gill
    Commented Jan 26, 2017 at 9:28
  • $\begingroup$ Or $Z_{31}$ in $L_5(2)$ of $Z_{127}$ in $ L_7(2)$ or $Z_{8191}$ in $L_{13}(2)$, etc., or $Z_{1093}$ in $L_7(3)$. $\endgroup$
    – Derek Holt
    Commented Jan 26, 2017 at 9:35
  • $\begingroup$ @Professors Holt and Gill. Can you help me to find a complete classification of such groups? $\endgroup$
    – H.Shahsavari
    Commented Jan 26, 2017 at 9:51
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    $\begingroup$ Or examples of form $Z_p \rtimes Z_{q^2}$ with nontrivial action but with $Z_q$ centralizing $Z_p$. I think you could prove that $G$ either has one of a limited list of easy structures or $G$ is simple. Either $|G|=p^3$ or a Sylow $p$-subgroup of $G$ has order $p$ or $p^2$ and hence is abelian. If it is self-normalizing then you can apply Burnside's Transfer Theorem to get a normal $p$-complement. That's my contribution! $\endgroup$
    – Derek Holt
    Commented Jan 26, 2017 at 11:35

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