= Exactly two composition series for a simple group times a prime cyclic group
{title2=$G=S\times C_p,\quad S\text{ nonabelian simple}$}
The only nontrivial proper <normal subgroups> of this <direct product of groups> are its two factors. Indeed a <normal subgroup> meeting $S\times1$ nontrivially contains it by the <simple group> property. Otherwise each of its commutators with $S\times1$ lies in that trivial intersection. Its first coordinates therefore lie in $Z(S)=1$, so it is contained in $1\times C_p$. Both factors are simple and each quotient by one factor is isomorphic to the other. Hence precisely two <composition series> occur, one through each factor. This is compatible with the <Jordan–Hölder theorem>, since the two series have the same simple factors in opposite orders. For $S=A_5,p=2$, the <group> has order $120$.
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