Answer to Question #145960 in Abstract Algebra for J

Question #145960
Assume that (G,*) is a group, and that H={a is an element of G: a=a^-1 }. H is a subgroup of G. Prove that H is a normal subgroup of G
1
Expert's answer
2020-11-24T05:25:31-0500

Let "H=\\{a\\in G:\\ a=a^{-1}\\}" be a subgroup of a group "(G,*)". To prove that "H" is a normal subgroup of "(G,*)", it is sufficient to prove that "g*a*g^{-1}\\in H" for all "a\\in H, g\\in G." Let "a\\in H, g\\in G" be arbitrary. Then "a^{-1}=a", and consequently, "(g*a*g^{-1})^{-1}=(g^{-1})^{-1}*a^{-1}*g^{-1}=g*a*g^{-1}". We conclude that "(g*a*g^{-1})^{-1}=g*a*g^{-1}", and therefore, "g*a*g^{-1}\\in H". Thus "H" is a normal subgroup of "(G,*)".


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