Q: Is every finite abelian group is cyclic?

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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.

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.

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.

The order of a group is the same as its cardinality - i.e. the number of elements the set contains. The order of a particular element is the order of the (cyclic) group generated by that element - i.e. the order of the group {...a-4, a-3, a-2, a-1, e, a, a2, a3, a4...}. If these powers do not go on forever, it will have a finite order; otherwise the order will be infinite.

Related questions

every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.

No.

By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.

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.

Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.

No.

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.

Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.

An abelian group is a group in which ab = ba for all members a and b of the group.

yes form cayleys theorem . every group is isomorphic to groups of permutation and finite groups are not an exception.

Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a

The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24