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In this book I am reading, a group is defined as:

A group is an ordered pair $(G,X)$ where $G$ is a set and $X$ is a binary operation on $G$ satisfying the following axioms:

i) $X$ is associative.

ii) $G$ contains an identity.

iii) Each element of $G$ has an inverse.

How does this definition imply that $G$ is closed under this operation?

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  • $\begingroup$ It's important that it is implicit in this def'n that each element has a double-sided inverse.That is, with identity 1, for any x in G there exists y with xy=yx=1. $\endgroup$ – DanielWainfleet Dec 10 '15 at 23:08
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"$X$ is a binary operation on $G$" is a shorthand for "$X$ is a map $G\times G\to G$".

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  • $\begingroup$ Oh of course. Thanks. $\endgroup$ – J.Gudal Dec 10 '15 at 21:51
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That $G$ is closed in hidden in the sentence "... $X$ is a binary operation on $G$ ..."

Saying this is saying that $X$ maps from $G\times G$ into $G$.

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