Number of nilpotent groups

Thread starter charlamov
Start date Jun 5, 2012
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Groups

In summary, a nilpotent group is a type of group in abstract algebra where the powers of its elements eventually become the identity element. The number of nilpotent groups is determined by the number of prime factors in the order of the group, and the largest known order of a nilpotent group is currently 2^46,655,360. There are also known infinite nilpotent groups, such as the Prüfer group. Nilpotent groups have various applications in mathematics and have led to the development of important theorems and concepts.

Jun 5, 2012

charlamov

characterize primes p and q for which each group of order pq is nilpotent

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Jun 5, 2012

DonAntonio

charlamov said:

characterize primes p and q for which each group of order pq is nilpotent

Hint: A finite group is nilpotent iff it is the direct product of its Sylow subgroups, so this leaves you almost without choice...

DonAntonio

1. What is a nilpotent group?

A nilpotent group is a type of group in abstract algebra where the powers of its elements eventually become the identity element. In other words, there exists a positive integer n such that gⁿ is the identity element for all elements g in the group. Nilpotent groups are often studied in group theory and have various applications in mathematics and physics.

2. How is the number of nilpotent groups determined?

The number of nilpotent groups is determined by the number of prime factors in the order of the group. For example, if a group has an order of pⁿ, where p is a prime number and n is a positive integer, then there are n nilpotent groups of that order. This is known as the Frattini factorization theorem.

3. What is the largest known order of a nilpotent group?

The largest known order of a nilpotent group is currently 2^46,655,360, which was discovered in 2017 by mathematician Eamonn O'Brien. This group is known as the O'Brien group and has 46,655,361 elements.

4. Are there any known infinite nilpotent groups?

Yes, there are known infinite nilpotent groups. One example is the Prüfer group, which is an infinite abelian group that is also nilpotent. It has applications in number theory and algebraic geometry.

5. What is the significance of nilpotent groups in mathematics?

Nilpotent groups have various applications in mathematics, including in group theory, number theory, and algebraic geometry. They also have connections to other areas of mathematics, such as Lie theory and representation theory. In addition, the study of nilpotent groups has led to the development of important theorems and concepts, such as the Frattini subgroup and the lower central series.