Group Theory : Cayley Tables, Cyclic Groups and Isomorphisms

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• I have the following cayley tables (which is in modulo 9)

determine the order of each element . Prove that G is a cyclic group.

• Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer.

• Let p be a prime number and G a group of order with identity element e. let and be a subgroup of G. prove that U is cyclic.

Solution:
1) We denote <a> the cyclic (sub)group generated by element (a), we ...