Is Cyclic Group Abelian?

Sophia Al-Mansoor
Sophia Al-Mansoor
• • 6 min read

All Cyclic Groups Are Abelian, but an Abelian Group Is Not Necessarily Cyclic. All Subgroups of an Abelian Group Are Normal. in an Abelian Group, Each Element...

Is Cyclic Group Abelian?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator

group generator
is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements in the group. Cyclic groups can be generated as powers of a single generator.
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How do you prove a cyclic group is abelian?

Since G is cyclic, it is generated by some element, say a. Then xy=(am)(an) for some m,n∈Z. Writing out this product, using the associativty, and then recollecting terms by definition of powers we see xy=am+n. Similarly, yx=am+n so that G is abelian.

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Is there any group which is cyclic but not abelian?

G=Z6×Z2G=Z6×Z2 will do (where ZnZn denotes the cyclic group of order nn). As a direct product of cyclic (so abelian) groups, GG is again abelian. ... Since no element of GG has order 12,12, then GG is not cyclic.

Sophia Al-Mansoor

Sophia Al-Mansoor

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