TECHNION, FACULTY OF MATHEMATICS ALGEBRA SEMINAR SPEAKER: Avi Goren, Technion. TITLE: Measuring arguments for finite relation preserving groups . DATE: Thursday, April 19, 2012 TIME: 14:30 NOTE UNUSUAL TIME, (BECAUSE OF THE HOLOCAUST REMEMBRANCE DAY CEREMONY AT 13:00.) PLACE: Amado 814 Abstract: Let G be a finite group acting on a set S. Let R be a relation on the set S. G is called a relation preserving group acting on S with relation R if for each x,y in S for which xRy we have also (x^g)R(y^g) for every g in G. For example, if G is a finite group acting on itself by conjugation define for x and y in G the relation xRy if xy=yx. Obviousely the conjugation preserves the relation R. If A is a subset of S, define R(A)={x in S| xRa for every a in A}. If G preserves the relation R of S ,O1 and O2 are G-orbits of S and A is a subset of S, we show a connection between the relative part of A in O1, the relative part of R(A) in O2 and the relative part of R(O1) in O2. We'll demonstrate how choosing the right G,S and R leads to various interesting results. For example: 1)If G is a finite group , cl1 and cl2 are conjugacy classes , and A is a subset of G we show a connection between the relative part of A in cl1, the relative part of the centralizer C(A) in cl2 and the relative part of C(cl1) in cl2. 2)If G is a finite group , N is a normal subgroup and a,b are elements in G we'll show that the relative part of Na in the conjugacy class cl(b) is <= 0.5 unless cl(b) is included in Na. (Actually it can be shown that the size of the intersection of Na with cl(b) divides the size of cl(b)) 3)If G is a finite group, A is an abelian subgroup of G and S is the set of conjugate subgroups of A , we show an upper bound for the size of B={H<=G | H is conjugate to A and the intersection of H and A is a subset of G's Fitting subgroup} 4)Let G be a finite group. Let cl1, cl2 and cl3 be three (not necessary distinct ) conjugacy classes of G. We'll show an upper bound for amount of elements x in cl1 such that x=yz where y is in cl2 and z is in cl3. --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Eli Aljadeff <elialjadeff@gmail.com>