Composition series

A composition series is a finite subnormal chain whose successive quotients are simple.

The only nontrivial proper normal subgroups of this direct product of groups are its two factors. Indeed a normal subgroup meeting $S\times 1$ nontrivially contains it by the simple group property. Otherwise each of its commutators with $S\times 1$ 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 $120$.

The sign quotient of $S_5$ controls its cyclic composition factors, while its unique nontrivial proper normal subgroup $A_5$ supplies the nonabelian factor. There are four composition chains in $C_2\times S_5$: the three maximal normal subgroups are the kernels of the nonzero maps to $C_2$, and only $C_2\times A_5$ has two choices for its next term. In $S_5\times S_5$ there are eight chains: six pass through $A_5\times A_5$ and one of its two simple direct factors; two instead pass through a single $S_5$ direct factor. The equal-sign subgroup gives two of the six chains. It has no quotient $A_5$, since its odd diagonal element acts by an outer automorphism on either alternating factor.

The composition length of a finite-length module or group is the number of simple factors in a composition series. The Jordan–Hölder theorem makes this independent of the chosen composition series.

A Jordan–Hölder factor is a simple quotient in a composition series. The Jordan–Hölder theorem says that the multiset of such factors is independent of the chosen composition series.

Any two composition series of a finite group or finite-length module have the same length and the same simple factors up to permutation and isomorphism.