Question

Let H be a normal subgroup of a group G, and let m = (G : H). Show thata^(m)inHfor every a in G

Abstract algebra
ANSWERED
asked 2021-02-26

Let H be a normal subgroup of a group G, and let \(m = (G : H)\). Show that
\(a^{m} \in H\)
for every \(a \in G\)

Expert Answers (1)

2021-02-27

Since H is a normal group of G and \(m=(G:H)\) then we have that the order of \(G/H\) is m. Therefore for every a in G we gave that \((aH)^m =H\), since the order of every element divides the order of the group,which implies \(a^m \in H\)
Result
Follows from the fact the assumptions imply \(|G/H|=m\)

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