Probabilistically nilpotent groups SHALEV, ANER
Proceedings of the American Mathematical Society,
04/2018, Volume:
146, Issue:
4
Journal Article
Peer reviewed
Open access
We show that, for a finitely generated residually finite group \Gamma , the word x_1, \ldots , x_k is a probabilistic identity of \Gamma if and only if \Gamma has a finite index subgroup which is ...nilpotent of class less than k. Related results, generalizations and problems are also discussed.
Almost PI algebras are PI Larsen, Michael; Shalev, Aner
Proceedings of the American Mathematical Society,
04/2022, Volume:
150, Issue:
4
Journal Article
Peer reviewed
Open access
We define the notion of an almost polynomial identity of an associative algebra R, and show that its existence implies the existence of an actual polynomial identity of R. A similar result is also ...obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras.
In the past few decades there has been considerable interest in word maps on groups with emphasis on (non-abelian) finite simple groups. Various asymptotic results (holding for sufficiently large ...groups) have been obtained. More recently non-asymptotic results (holding for all finite simple groups) emerged, with emphasis on particular words (commutators and certain power words) which are not an identity of any finite simple group. In this paper we initiate a systematic study of
all
words with the above property. In particular, we show that, if
w
1
,
…
,
w
6
are words which are not an identity of any (non-abelian) finite simple group, then
w
1
(
G
)
w
2
(
G
)
…
w
6
(
G
)
=
G
for
all
(non-abelian) finite simple groups
G
. Consequently, for every word
w
, either
w
(
G
)
6
=
G
for all finite simple groups, or
w
(
G
)
=
1
for some finite simple group. These theorems follow from more general results we obtain on
characteristic collections
of finite groups and their covering numbers, which are of independent interest and have additional applications.
McKay graphs for alternating and classical groups Liebeck, Martin W.; Shalev, Aner; Tiep, Pham Huu
Transactions of the American Mathematical Society,
08/2021, Volume:
374, Issue:
8
Journal Article
Peer reviewed
Open access
Let G be a finite group, and \alpha a nontrivial character of G. The McKay graph \mathcal {M}(G,\alpha ) has the irreducible characters of G as vertices, with an edge from \chi _1 to \chi _2 if \chi ..._2 is a constituent of \alpha \chi _1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G = \mathsf {A}_n, we prove a conjecture made in another work by the authors: there is an absolute constant C such that \mathrm {diam} {\mathcal M}(G,\alpha ) \le C\frac {\log |G|}{\log \alpha (1)} for all nontrivial irreducible characters \alpha of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a product \chi _1\chi _2\cdots \chi _l of irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents.
On the distribution of values of certain word maps Larsen, Michael; Shalev, Aner
Transactions of the American Mathematical Society,
March 1, 2016, 20160301, 2016-3-00, Volume:
368, Issue:
3
Journal Article
Peer reviewed
Open access
We prove that, for any positive integers m, n (x,y) \mapsto x^my^n obtained in 2009. Along the way we obtain results of independent interest on fibers of word maps and on character values.
Let
G
be a finite group, and
α
a nontrivial character of
G
. The McKay graph
ℳ
(
G
,
α
)
has the irreducible characters of
G
as vertices, with an edge from
χ
1
to
χ
2
if
χ
2
is a constituent of
αχ
1
.... We study the diameters of McKay graphs for simple groups
G
of Lie type. We show that for any
α
, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for
G
= PSL
n
(
q
) or PSU
n
(
q
).
The depth of a finite simple group BURNESS, TIMOTHY C.; LIEBECK, MARTIN W.; SHALEV, ANER
Proceedings of the American Mathematical Society,
06/2018, Volume:
146, Issue:
6
Journal Article
Peer reviewed
Open access
We introduce the notion of the depth of a finite group G, defined as the minimal length of an unrefinable chain of subgroups from G to the trivial subgroup. In this paper we investigate the depth of ...(non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.
The length and depth of compact Lie groups Burness, Timothy C.; Liebeck, Martin W.; Shalev, Aner
Mathematische Zeitschrift,
04/2020, Volume:
294, Issue:
3-4
Journal Article
Peer reviewed
Open access
Let
G
be a connected Lie group. An unrefinable chain of
G
is defined to be a chain of subgroups
G
=
G
0
>
G
1
>
⋯
>
G
t
=
1
, where each
G
i
is a maximal connected subgroup of
G
i
-
1
. In this ...paper, we introduce the notion of the length (respectively, depth) of
G
, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups
G
. We obtain best possible bounds on the length of
G
in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on
dim
G
′
in terms of the chain difference of
G
, which is its length minus its depth.
Let G be a finite simple group. We show that the commutator map \alpha :G \times G \rightarrow G is almost equidistributed as |G| \rightarrow \infty . This somewhat surprising result has many ...applications. It shows that a for a subset X \subseteq G we have \alpha ^{-1}(X)/|G|^2 = |X|/|G| + o(1), namely \alpha is almost measure preserving. From this we deduce that almost all elements g \in G can be expressed as commutators g = x,y where x,y generate G. \par This enables us to solve some open problems regarding T-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T-systems in G with two generators tends to infinity as |G| \rightarrow \infty . This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. \par Some of our results apply for more general finite groups and more general word maps. \par Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function \zeta ^G(s) = \sum _{\chi \in \operatorname {Irr}(G)}\chi (1)^{-s} plays a key role in the proofs.
Rapid growth in finite simple groups LIEBECK, MARTIN W.; SCHUL, GILI; SHALEV, ANER
Transactions of the American Mathematical Society,
12/2017, Volume:
369, Issue:
12
Journal Article
Peer reviewed
Open access
We show that small normal subsets A of finite simple groups grow very rapidly; namely, \vert A^2\vert \ge \vert A\vert^{2-\epsilon }, where \epsilon >0 is arbitrarily small. Extensions, consequences, ...and a rapid growth result for simple algebraic groups are also given.
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