the divisibility graph $mathscr{d}(g)$ for a finite group $g$ is a graph with vertex set ${rm cs}(g)setminus{1}$‎ ‎where ${rm cs}(g)$ is the set of conjugacy class sizes of $g$‎. ‎two vertices $a$ and $b$ are adjacent whenever $a$ divides‎ ‎$b$ or $b$ divides $a$‎. ‎in this paper we will find the number of connected components of $mathscr{d}(g)$ where $g$ is a‎ ‎simple zassenhaus group or an sp...

m ƒ qn‚ r , and 0 r < jnj. Not surprisingly, q is the quotient and r is the remainder when m is divided by n.1 We say that n divides m, or n is a divisor of m, when the remainder is zero. The notation n jm is used to signify that n divides m. An integer p is a prime number if 1 < jpj and the only divisors of p are 1 and p. We write a b „mod n… if b a is divisible by n.2 This notion, called cong...

(4) Let a > 1 be an integer. Prove that for any positive integers n, d that d divides n if and only if a − 1 divides a − 1 (cf. the previous exercise). Conclude in particular that Fpd ⊆ Fpn if and only if p divides n. Proof. The proof from the previous problem shows that d divides n if and only if a − 1 divides a − 1; the only change is that in the last step we argue by size rather than degree ...

LetN be a square free integer, and let A be an optimal elliptic curve over Q of conductor N . We prove that if A has a rational torsion point of prime order r such that r does not divide 6N , then r divides the order of the cuspidal subgroup of J0(N).

A classical conjecture of Erdős and Sós asks to determine the Turán number a tree. We consider variants this problem in settings hypergraphs multi-hypergraphs. In particular, for all k r, with r≥k(k−2), we show that any r-uniform hypergraph H more than n(k−1)r+1 hyperedges contains Berge copy tree edges different from k-edge star. This bound is sharp when r+1 divides n such values extremal hype...

[n is an arbitrary natural number, r is an arbitrary (nonzero) real quantity) gives & positive integer k. Since both r and k turn out to be Fibonacci number ratios, the results established in this paper can be viewed as a particular kind of Fibonacci identities that are believed to be new [see (4.7) and (4.8)]. Throughout the paper we shall make use of the following properties of the Fibonacci ...

Let G be a group with identity e. R G-graded commutative ring and M graded R-module. We introduce the concept of Ie-prime submodule as generalization prime for I =?g?G Ig fixed ideal R. give number results concerning this class submodules their homogeneous components. A proper N is said to if whenever rg ? h(R) mh h(M) rgmh IeN, then either (N :R M) or N.

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x− r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) ...