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charlamov
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characterize primes p and q for which each group of order pq is nilpotent
charlamov said:characterize primes p and q for which each group of order pq is nilpotent
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.
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.
The largest known order of a nilpotent group is currently 246,655,360, 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.
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.
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.