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G is a generalized inverse of a if and only if aga=a.g is said to be reflexive if and only if gag=g I'm sure i must be wrong, but my approach was to again show that $\sim$ is an equivalence relation. I was trying to solve the problem

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If a is a matrix and g be it's generalized invers. My confusion lies in the fact that they appear to be the same question This is stated without proof in dummit and foote.

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We have a group $g$ where $a$ is an element of $g$ Then we have a set $z (a) = \ {g\in g Ga = ag\}$ called the centralizer of $a$ If i have an $x\in z (a)$, how.

Prove that $\forall u,v\in g$, $uv\sim vu$