Finite abelian groups with elements of certain orders

Give examples of finite abelian groups in which all elements (except the identity element) are of the same order.

Solution Preview

Suppose that n > 1, and we are asked to name an abelian group all of whose elements are of order n. Technically, no such group exists, since the identity element of every group (abelian or otherwise) is of order 1.

So suppose we try to find finite abelian groups such that all the elements except the identity element are of the same order.

One important class of abelian groups consists of the cyclic groups (the groups that can be generated from a single element).

If n > 1 and n is the order of the cyclic group of order n, then every element is of order m for some ...

Solution Summary

Examples are given of finite abelian groups in which all elements but the identity element are of the same order. The reason why they have the same order is explained.

... b) = (a^2)(b^2) = (ab)^2 [Since G is Abelian] = f(ab ... 3: Prove that f is surjective (onto): Since G is finite, and f ... 2. Homomorphism: Let G and H be two groups. ...

... the group has four elements, show it must be abelian. Solution:- Let , where is the identity element of . Order of . Order of any element of a finite group is ...

... If p is a prime, any two groups of order p are ... of a group G is a normal subgroup, then G is abelian. ... Prove that a finite p-group G is simple if and only if the ...

... O(G ) = 5 , a prime , then non of the groups of order 4 is not ... But it is not abelian. ... O (a ) If O(a ) = ∞ then () () O a −1 ≤ ∞ ⇒ O a −1 is finite. ...

... the order of any element g of a finite group (ie the ... Free_group 5. The direct product of solvable groups is solvable ... is normal in G and G /G is abelian, for any ...

... Cauchy's Theorem, Order, Abelian Groups, Non-Abelian Groups, Isomorphisms and ... Sym(S). Thus any non-Abelian group of order... H. Assuming that G is finite and let ...

... Write down all direct products of cyclic groups that could be ... Since G is a finite group, so | f (G) |=| G |= n ... G is an abelian group with order 16 = 2 4 . So ...

... ii) We are given an abelian group G. We wish to ... G consisting of all elements of G of finite order is a ... that H is closed under inversion and group multiplication ...

... it implies that the number of cosets of H is finite. ... 24)} is a normal subgroup of A4 and H is abelian. The quotient group GH has order | G | 12 = 3 , which is ...

... Prove that [O:Of]= f (index as additive abelian groups). ... of 0 containing the identity and having finite index f in 0 (as additive abelian group) is equal to ...