Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images

Details Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images

2007-08-16

arxiv.org/abs/0708.2280v1

Mathematics

Group Theory

Scientific paper

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that every 2-generator $E$-group is abelian and that all 3-generator $E$-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator $E$-group is abelian. We conjecture that every finite 3-generator $E$-group should be abelian. Moreover we show that the minimum order of a non-abelian $pE$-group is $p^8$ for any odd prime number $p$ and this order is $2^7$ for $p=2$. Some of these results are proved for a class wider than the class of $E$-groups.

Affiliated with

Also associated with

No associations

Rating

Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-5557