We investigate various strong notions of rigidity for Souslin trees, separating them under ♦ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ♦ that there is a group whose automorphism tower is highly malleable by forcing.

It is common in mathematics for a subject to have its local and global aspects; such is the case in group theory. For example, the structure and embedding of subgroups of a group G may be usefully thought of as part of the local structure of G while the normal subgroups, quotient groups and conjugacy classes are relevant to the global structure of G. Furthermore, the connections between local a...

We are now ready to apply the theory of group actions we studied in the last section to study the general structure of finite groups. A key role is played by the p-subgroups of a group. We will see that the Sylow theory will give us a way to study a group “a prime at a time”. First we record a very important special case of group actions: Theorem 1.1 (p-group Actions). Let p be a prime. If P is...

We give simple arithmetic conditions that force the Sylow p -subgroup of critical group a strongly regular graph to take specific form. These depend only on parameters ( v , k λ μ ) under consideration. many examples, including how theory can be used compute Conway's 99-graph and an elementary argument no s r g 28 9 0 4 exists.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

The Algebraicity Conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been the so-called Borovik program of transferring methods from finite group theory. This program has led t...

Abstract A classical result of Baer states that a finite group G which is the product two normal supersoluble subgroups if and only ʹ nilpotent. In this article, we show = AB (respectively, w -supersoluble) B , in permutes with every maximal subgroup each Sylow then -supersoluble), provided We also investigate products defined above when $ A\cap B=1 obtain more general results.

Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups an...

the goal of present study was to compare the performance of adhd and normal children on planning tests. 45 children with adhd aged between 7 to 12 were selected in mashhad city through convenience and randomized sampling. we used snap، cbcl، trf, csi-4 and clinical interview for diagnosis of adhd and put them into adhd-i, adhd-h, adhd-c subtypes. each subtype comprised 15 children with adhd. th...