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The powers P n are normal subgroups of E(p), and the example groups are E(p,n) = E(p)/P n. E(p,n) has order p n+1 and nilpotency class n, so is a p-group of maximal class. When p = 2, E(2,n) is the dihedral group of order 2 n. When p is odd, both W(2) and E(p,p) are irregular groups of maximal class and order p p+1, but are not isomorphic.
A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch. 12.4), (Huppert 1967, Kap.III §10) conditions are satisfied: For every a, b in G, there is a c in the derived subgroup H ′ of the subgroup H of G generated by a and b, such that a p · b p = (ab) p · c p.
More generally, a subgroup of index p where p is the smallest prime factor of the order of G (if G is finite) is necessarily normal, as the index of N divides p! and thus must equal p, having no other prime factors. For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius ...
For n > 1, the maximal nilpotency class of a group of order p n is n - 1 (for example, a group of order p 2 is abelian). The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups. Furthermore, every finite nilpotent group is the direct product of p-groups. [5]
Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript notation means the n-fold direct product of groups. [2] In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. [4] (Note that in the finite case the direct product and direct sum coincide ...
The semidirect product of a cyclic group of order p 2 by a cyclic group of order p acting non-trivially on it. This group has exponent p 2. If n is a positive integer there are two extraspecial groups of order p 1+2n, which for p odd are given by The central product of n extraspecial groups of order p 3, all of exponent p.
The p-group generation algorithm by M. F. Newman [1] and E. A. O'Brien [2] [3] is a recursive process for constructing the descendant tree of an assigned finite p-group which is taken as the root of the tree.
This U is a pro-p group. In fact the p-adic analytic groups mentioned above can all be found as closed subgroups of for some integer n, Any finite p-group is also a pro-p-group (with respect to the constant inverse system). Fact: A finite homomorphic image of a pro-p group is a p-group. (due to J.P. Serre)