Direct factor implies normal
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., direct factor) must also satisfy the second subgroup property (i.e., normal subgroup)
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Statement
Suppose is an internal direct product of subgroups
and
, so that both
and
are direct factors of
. Then, both
and
are normal subgroups of
.
Intermediate properties
Proof
The proof is direct from the definition.