# A set of bijections

Let $A$ is a (possibly infinite) set.

Let $G$ is a group of functions (more precisely, bijections) on $A$ with function composition.

How to call such a group?

• a group of permutations of $A$;
• a group of bijections on $A$;
• whatever.
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I think that the usual notation is permutations, but the other two are also appropriate ;) –  N. S. Oct 13 '12 at 18:39
Also the usual way to write the first two sentences is "Let A be a..." –  Idan Oct 13 '12 at 18:52

I think it's typical to say:

"$G$ is a permutation group on $A$".

If we do a Google Scholar search for "G is a permutation group on" (in quotes), it comes up with numerous respectable examples of this phrase being used.

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This group G is commonly called "the symmetric group on A", especially if A is finite. A shorthand notation is $S_A$.
Porton's group $G$ is a subgroup of $S_A$, not necessarily the whole group. –  Derek Holt Oct 13 '12 at 20:18