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Group of order 9

This is the question:

Consider small groups.

(i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3

(ii) List all groups of order at most 10 (up to isomorphism)

Solution Preview

(i) Proof:
First, I show that G is abelian if the order of G is 9. Since |G|=9=3^2 and 3 is a prime number, then G is a p-group. Any p-group has nontrivial center Z(G). Then Z(G) has order 3 or 9. If the order of Z(G) has order 3, then Z(G) is a cyclic group. We can assume ...

Solution Summary

This solution is comprised of a detailed explanation to show that a group of order 9 is isomorphic to Z9 or Z3 x Z3.

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