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 ...

Let G be a group (finite or infinite) and let a and b in G. Let o(a) represent the order of a. Suppose G is abelian and both a and b are of finite order. ...

... 2 Actually, we consider the following 5 finite groups of order... Third, we consider the two non-abelian groups. ... Hence all the 5 groups are not isomorphic to each ...

... 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. ...

... for all primes p. By the fundamental theorem of finite abelian groups [1], we know that G is the direct product of a finite number of cyclic groups, each of ...

... 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 ...

... a composite number Then G is a finite non-abelian simple group... O(G)/O(N(K)) > 5 For G is a finite group. ... This gives 15.(23-1) = 15.7 = 105 elements of order 2i ...

... 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 ...