P-GROUP

In mathematics, given a prime number ''p'', a '''p''-group' is a periodic group in which each element has a power of ''p'' as its order. That is, for each element ''g'' of the group, there exists a nonnegative integer ''n'' such that ''g'' to the power ''pⁿ'' is equal to the identity element. Such groups are also called 'primary'.
If ''G'' is finite, this is equivalent to requiring that the order of ''G'' (the number of its elements) itself be a power of ''p''. Quite a lot is known about the structure of finite ''p''-groups. One of the first standard results using the class equation is that the center of a non-trivial finite ''p''-group cannot be the trivial subgroup. A finite ''p''-group with order ''pⁿ'' contains subgroups of order ''pⁱ'' with 0 ≤ ''i'' ≤ ''n''. More generally, every finite ''p''-group is nilpotent, and therefore solvable.
''p''-groups of the same order are not necessarily isomorphic; for example, the cyclic group ''C''₄ and the Klein group ''V''₄ are both 2-groups of order 4, but they are not isomorphic. Nor need a ''p''-group be abelian; the dihedral group ''Dih''₄ of order 8 is a non-abelian 2-group. However, every group of order ''p''² is abelian.
In an asymptotic sense, almost all finite groups are ''p''-groups. In
fact, almost all finite groups are 2-groups: the fraction of isomorphism classes of 2-groups
among isomorphism classes of groups of order at most ''n'' tends to 1 as ''n'' tends to infinity. For instance, of the 49 910 529 484 different groups of order at most 2000, 49 487 365 422, or just over 99%, are 2-groups of order 1024.^[1]
Every non-trivial finite group contains a subgroup which is a non-trivial ''p''-group. The details are described in the Sylow theorems.
For an infinite example, see Prüfer group.

Contents

See also

References

See also

★ nilpotent group

★ Sylow subgroup

★ Prüfer rank

★ Extra special group

References

1. Hans Ulrich Besche, Bettina Eick, and Eamonn O'Brien. A millennium project: constructing Small Groups
Internat. J. Algebra Comput. 12, 623 - 644 (2002).