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Suppose I have some set $G=\{g_1,g_2,g_3,g_4\}$ and the multiplication operation "$\cdot$". In Mathematical Methods for Physics and Engineering by Riley, Hobson, Bence, the authors write:

A group is a set of elements $\{X,Y,...\}$, together with a rule for combining them...for which the following conditions must be satisfied:

  1. For every pair of elements $X,Y$ that belongs to the set $X \cdot Y$ must also belong to the set.

In my example, does that also mean that $g_1 \cdot g_1$ must be in the set in order for it to be a group or is it only required for different elements, i.e $g_i \cdot g_j$?

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  • $\begingroup$ Yes, $g_1\cdot g_1$ must be in the set. $\endgroup$ – almagest Jun 2 '16 at 15:17
  • $\begingroup$ @almagest Thanks! $\endgroup$ – bluemoon Jun 2 '16 at 15:19
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$g_1, g_1$ is also a pair of elements that belongs to $G$, so yes, $g_1 \cdot g_1$ must be in the set.

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