Let $G$ be a finite group and $p$ be a prime. Suppose that every minimal normal subgroup of $G$ is isomorphic to $\mathbb{Z}_{p}$. What possible structures does $G$ have?
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1 Answer
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Some easy observations (a bit too long for a comment):
Every $p$-group has this property, as does every quasisimple group whose centre is a non-trivial $p$-group. Conversely if $G$ has this property, the generalized Fitting subgroup has the form $O_p(G)E(G)$ and the centre of every component is a non-trivial $p$-group. So given a component $Q$ of $G$, then $Q/Z(Q)$ has Schur multiplier divisible by $p$; given the classification of finite simple groups, for $p > 3$ I think all you have left are some of the special linear and unitary groups. For $p \in \{2,3\}$ there is a longer list of possible components.
Given a group $G$ with generalized Fitting subgroup as described, then every minimal normal subgroup is contained in the elementary abelian group $M := \Omega_1(Z(O_p(G)))$, so it will come down to the structure of $M$ as a $G$-module.
