Demonstrate that for the cyclic group (mathrm{C}_{4}) with multiplication table given in Example 2.15 , the subgroup (H=left{e, a^{2} ight}) is an abelian invariant subgroup.

Demonstrate that for the cyclic group \(\mathrm{C}_{4}\) with multiplication table given in Example 2.15 , the subgroup \(H=\left\{e, a^{2}\right\}\) is an abelian invariant subgroup.

Data from Example  2.15

Example 2.15 Consider the cyclic group of order four, C4 = {e, a, a, a), where e = a+ defines the cyclic condition, with a multiplication table C4 e a a a e e a a a a a a a e a a a e a a a e a a As shown in Problem 2.5, the subgroup H = {e, a) is identical to all of its conjugate subgroups H' = {ghg; g G, he H), and is an (abelian) invariant subgroup of C4.