(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [tex](G, \circ)[/tex] is a group. Define an operation [tex]\star[/tex] on [tex]G[/tex] by [tex]a \star b = b \circ a[/tex] for all [tex]a,b \in G[/tex]. Show that [tex](G,\star)[/tex] is a group.

3. The attempt at a solution

So, I have to show that [tex](G,\star)[/tex] satisfies the associativity, identity, inverse and closure conditions:

Associativity: Let a,b,c be elements in G.

[tex](a \star b) \star c = (b \circ a) \star c[/tex]

[tex]= c \circ (b \circ a) = (c \circ b) \circ a[/tex]

[tex]= a \star (b \star c)[/tex].

Identity: [tex](a \star e) = (e \circ a) = (a \circ e)=(e \star a) = a[/tex].

Inverse: y is an inverse of a then [tex](a \star y)= (y \circ a) = (y \star a) = e[/tex].

Are these correct so far??

And for the closure, do I just need to show that any element of [tex](G,\star)[/tex] is in [tex]G[/tex]? I know that [tex](a \star b)=(b \circ a)[/tex] and [tex](b \circ a) \in (G, \circ)[/tex] so is that all I need to say?

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# Homework Help: Opposite groups - show it is a group

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