21. In a group ( G ), let ( C ) be the center and ( C_{a} ) be the centralizer of ( a ). Prove Theorem 3 by showing the following. (a) ( C_{a} ) is a subgroup in ( G ) for all ( a ) in ( G ). (b) ( C ) is a subgroup in ( C_{a} ) for all ( a ) in ( G ). (c) An element ( g ) is in ( C ) if and only if ( g ) is in ( C_{

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21. In a group  ( G  ), let  ( C  ) be the center and  ( C_{a}  ) be the centralizer of  ( a  ). Prove Theorem 3 by showing the following. (a)  ( C_{a}  ) is a subgroup in  ( G  ) for all  ( a  ) in  ( G  ). (b)  ( C  ) is a subgroup in  ( C_{a}  ) for all  ( a  ) in  ( G  ). (c) An element  ( g  ) is in  ( C  ) if and only if  ( g  ) is in  ( C_{a}  ) for every  ( a  ) in  ( G  ). (d) An element  ( g  ) is in  ( C  ) if and only if  ( C_{g}=G  ).  3 Centralizers and the Center In a group  ( G  ), let  ( C  ) be the center and  ( C_{a}  ) be the centralizer of  ( a  ). Then: (a)  ( C_{a}  ) is a subgroup in  ( G  ) for every  ( a  ) in  ( G  ). (b)  ( C  ) is a subgroup in  ( C_{a}  ) for every  ( a  ) in  ( G  ). (c) An element  ( g  ) is in  ( C  ) if and only if  ( g  ) is in  ( C_{a}  ) for every  ( a  ) in  ( G  ). (d) An element  ( g  ) is in  ( C  ) if and only if  ( C_{g}=G  ). The proof of this result is left to the reader as Problem 21 of this section.

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Solution for -  21. In a group  ( G  ), let  ( C  ) be the center and  ( C_{a}  ) be the centralizer of  ( a  ). Pr

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(Solved): 21. In a group \( G \), let \( C \) be the center and \( C_{a} \) be the centralizer of \( a \). Pr ...

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