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In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is termed malnormal if for any x {\displaystyle x} in G {\displaystyle G} but not in H {\displaystyle H} , H {\displaystyle H} and x H x − 1 {\displaystyle xHx^{-1}} intersect in the identity element.
Some facts about malnormality:
When G is finite, a malnormal subgroup H distinct from 1 and G is called a "Frobenius complement". The set N of elements of G which are, either equal to 1, or non-conjugate to anyelement of H, is a normal subgroup of G, called the "Frobenius kernel", and G is the semi-direct product of H and N.