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- If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.
- Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.