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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.)

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.

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Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <techm@math.technion.ac.il>