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normal subgroup If H is a subgroup of a group G, and for any element, x of G, the left and right cosets of H are equal, then H is a normal subgroup. This is denoted H ⊳ G. Equivalently, H is closed under conjugation; that is, for every x in G and h in H, then x–1hx is in H. In an abelian group, all subgroups are normal, and in general G and {e} are normal subgroups of G. The normal subgroups are precisely those subgroups that form a well-defined quotient group. See also simple group.
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