The MOR cryptosystem [9] is a natural generalization of the El-Gamal cryptosystem to non-abelian groups. Using a p-group, a cryptosystem was built in [4]. It seems resoanable to assume the cryptosystem is as secure as the El-Gamal cryptosystem over finite fields. A natural question arises can one make a better cryptosystem using p-groups? In this paper we show that the answer is no.

Suppose that  is a finite group. Then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . Suppose that . Then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . In this paper, we prove that if  is a group such that , where , then . Here  denotes the family of Suzuk...

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

One of the important problems in group theory is characterization of a group by a given property, that is, to prove there exist only one group with a given property. Let  be a finite group. We denote by  the largest order of elements of . In this paper, we prove that some Suzuki groups are characterizable by order and the largest order of elements. In fact, we prove that if  is a group with  an...