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## Main Question or Discussion Point

https://imgur.com/a/oSioYel

I am trying to understand this proof, but am tripped up on the part that says "Consider the action of ##G## on ##\operatorname{Syl}_2(G)## by conjugation." My question is, how is this a well-defined action if ##\operatorname{Syl}_2(G)## is not normal? Isn't this action by conjugation defined only on subgroups that are normal?

I am trying to understand this proof, but am tripped up on the part that says "Consider the action of ##G## on ##\operatorname{Syl}_2(G)## by conjugation." My question is, how is this a well-defined action if ##\operatorname{Syl}_2(G)## is not normal? Isn't this action by conjugation defined only on subgroups that are normal?