Class numbers of p-groups of a given order

Classification of solvable groups with a given property

In this paper we classify all finite solvable groups satisfying the following property P5: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.

Finiteness of Class Numbers for Algebraic Groups

Let G be an affine group scheme of finite type over a global field F . (We do not assume G to be reductive or smooth or connected.) Let AF denote the locally compact adele ring of F , S be a finite non-empty set of places of F that contains the archimedean places, and AF the factor ring of adeles with vanishing component along S; for FS = ∏ v∈S Fv, we have AF = FS ×AF . Recall that if X is a fi...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations

groups of order $p^8$ and exponent $p$

we prove that for $p>7$ there are‎ ‎[‎ ‎p^{4}+2p^{3}+20p^{2}+147p+(3p+29)gcd (p-1,3)+5gcd (p-1,4)+1246‎ ‎] ‎groups of order $p^{8}$ with exponent $p$‎. ‎if $p$ is a group of order $p^{8}$‎ ‎and exponent $p$‎, ‎and if $p$ has class $c>1$ then $p$ is a descendant of $‎p/gamma _{c}(p)$‎. ‎for each group of exponent $p$ with order less than $‎p^{8} $ we calculate the number of descendants of o...

Finite P-groups of Class 2 Have Noninner Automorphisms of Order P

We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ(G) or Ω1(Z(G)) elementwise fixed.