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Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$be a prime. We prove that $p$ does not divide the conjugacy class size (index)of each $p$-regular element of prime power order $x\in A\cup B$ if and only if$G$ is $p$-decomposable, i.e. $G=O_p(G) \times O_{p'}(G)$.

On products of groups and indices not divisible by a given prime