Yes. Lets call the generator of the group z, then every element of the group can be written as zk for some k. Then the product of two elements is: zkzm=zk+m
Notice though that then zmzk=zm+k=zk+m=zkzm, so the group is indeed abelian.
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No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic
The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24
The set of integers, under addition.
Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
rectangle has inversion (180 deg rotation) hexagon has 60 deg ratation, cyclic group genterated is 60, 120, 180, 240, 300, 360=0 equilateral triangle has 120 deg rotation, cyclic group genterated is 120, 240, 360=0