Write down the elements of group under composition

Test 0: Is the operation associative? If not, then youβre done. π(π)

is not a group. If the operation is associative, then proceed toβ¦

Test 1: Does the set π(π)

contains an identity? If you have a candidate function in π(π), then what equations must it satisfy? Does it? If not, then π(π)

is not a group. If you do have an identity, then proceed toβ¦

Test 2: Does every element of π(π)

have an inverse? Given an arbitrary function in πβπ(π), can you write down its inverse πβ1βπ(π)? What equation must π and πβ1 satisfy? (Hint: you need the identity function.) If any function fails to have an inverse, then π(π)

is not a group. If every function does have an inverse, thenβ¦

Congratulations! Your set π(π)

is a group.

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