The seminar meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.
Organizer: Marcin Mazur
Is it true that a p-group of odd order can be generated by elements of the same order?
obtained by O'Brien, Scoppola and Vaughan-Lee.
We show that for a finite abelian group the set of autocommutators is equal to the autocommutator subgroup. With the help of GAP we obtain a group of order 64 in which not every element of the autocommutator subgroup is an autocommutator. This group is of minimal order with this property.
This talk will be a survey of the use of algebraic and combinatorial methods to study certain sets of piecewise polynomial functions (also known as splines) defined on a triangulated region of Rn. These functions are used extensively throughout applied mathematics and engineering, but they lend themselves to an interesting theoretical study as well. For example, the sets can be viewed as subrings of the Stanley-Reisner face ring of a simplicial complex.
A subgroup M of a group G is a permutable subgroup if for all subgroups X of G, we have MX=XM. It is then natural for us to define the notion of a permutably detectable group as a group G where for any direct product of finitely many copies of G, the only permutable subgroups isomorphic to G are the direct factors. In this talk, we will present some examples, conjectures, and results concerning finite permutably detectable groups.
Conjecture 1: For a given prime p, the elements of order dividing p in a group G always form a subgroup.
Obviously the conjecture is wrong. But what is the order of a minimal counterexample? Denoting with f(p) the order of a minimal counterexample, we have the following results.
Theorem 2: Let p be a prime. Then f(p)=p(p+1) if and only if p=2 or a Meresenne prime.
For odd primes, other than Mersenne primes the result is more complicated.
Theorem 3. Let p be an odd prime which is not a Mersenne prime, then f(p)=min( p(kp+1), |PSL(2,p)| ) where k is the smallest integer such that kp+1 s a prime power.
As can be easily verified, both cases occur and because of the erratic behavior of k, not much can be said about it. The following conjecture can easily be seen to be wrong.
Conjecture 4: For a given prime p, the Sylow p-subgroup of a group G is always normal in the group.
Denoting the order of a minimal counterexample to Conjecture 4 by n(p), Theorem 2 and 3 provide us with the following corollary.
Corollary 5: For any prime p, f(p)=n(p).
This is joint work with Luise-Charlotte Kappe and Michael Ward.
In this lecture I will present a structure theorem for an abstract finite simple group G. It is the theoretical basis for an algorithm constructing the simple target groups G from a presentation of the given group H. The structure theorem, Thompson's group order formula and my group order formula allow the calculation of the orders of the target groups G by using only data which can be derived from the presentation of H. Successful applications of these results like the uniqueness proof by Weller, Previtali and myself of the long standing problem of the uniqueness of the sporadic Thompson group will be mentioned. Furthermore, some serious open problems in the classification of finite simple groups will be discussed and documented.
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