Let $A$ and $B$ be two subgroups of the same group $G$. What does it mean for the subgroup $A$ to be normalized by the subgroup $B$?
normalized subgroup by another subgroup
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group-theory
definition
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2It means that $A \lhd \langle A,B \rangle$ is one way to say it. – 2011-07-29
1 Answers
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It means that for every $b\in B$, $A^b = \{b^{-1}ab\mid a\in A\} = A$. That is, that $B$ is a subgroup of the normalizer of $A$ in $G$.
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1@palio: In my previous example, $A$ normalizes $B$, of course; but it is also possible to have $AB=BA$ and yet have neither $A$ normalize $B$, nor $B$ normalize $A$. The statement is simply false. – 2011-07-29