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...

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the ‎order of $text{aut}(g)$‎.

In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3 -strong dimension 0. We also show that P/poly i.o. has p 3 -dimension 1/2, p 3 -strong dimension 1. Our results improve previous measure results of Lutz (1992) and dimension results of Hitchcock and Vinodchandran (2004).

The problem of counting unlabelled subtrees of a tree (that is, subtrees that are distinct up to isomorphism) is #P-complete, and hence equivalent in computational difficulty to evaluating the permanent of a 0,1-matrix.

Implications of a formal context (G,M, I) have a minimal implication basis, called Duquenne-Guigues basis or stem base. It is shown that the problem of deciding whether a set of attributes is a premise of the stem base is in coNP and determining the size of the stem base is polynomially Turing equivalent to a #P-complete problem.