Elements Of Prime Order Research Articles

Weights of Partitions and Character Zeros

We classify partitions which are of maximal $p$-weight for all odd primes $p$. As a consequence, we show that any non-linear irreducible character of the symmetric and alternating groups vanishes on some element of prime order.

On Cremona transformations of prime order

We prove that an automorphism of order 5 of the Del Pezzo surface S of degree 5 is conjugate through a birational map S ▪ P 2 to a linear automorphism of P 2 . This completes the classification of conjugacy classes of elements of prime order in the Cremona group. To cite this article: A. Beauville, J. Blanc, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

On automorphisms of strongly regular graphs with ,

The structure of fixed-point subgraphs of automorphisms of order 3 of strongly regular graphs with parameters is determined. Let be the automorphism group of a hypothetical strongly regular graph with parameters . Possible orders are found and the structure of fixed-point subgraphs is determined for elements of prime order in . The four-subgroups of are classified and the possible structure of the group is determined. A strengthening of a result of Nakagawa on the automorphism groups of strongly regular graphs with , is obtained.

Fixed point spaces in actions of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0, and let H be a maximal closed non-subspace subgroup of G. Given such a pair (G, H), we obtain a close to best possible upper bound for the ratio dim(xG ? H) / =dim xG, where x ? G is a semisimple or unipotent element of prime order. We apply this result to the study of fixed point spaces.

On planar Cremona maps of prime order

AbstractThis paper contains a new proof of the classification of prime order elements of Bir(ℙ2) up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

QUASIPRIMITIVE GROUPS WITH NO FIXED POINT FREE ELEMENTS OF PRIME ORDER

The paper determines all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order. All such groups are primitive and are wreath products in a product action involving $M_{11}$ in its action on 12 points. These groups are not 2-closed and so substantial progress is made towards asserting the truth of the polycirculant conjecture that every 2-closed transitive permutation group has a fixed point free element of prime order. All finite simple groups $T$ with a proper subgroup meeting every ${\rm Aut}(T)$ -conjugacy class of elements of $T$ of prime order are also determined.

A new notion of rank for finite supersolvable groups and free linear actions on products of spheres

For a finite supersolvable group G, we define the saw rank of G to be the minimum number of sections Gk i Gki1 of a cyclic normal series G⁄ such that Gk i Gki1 owns an element of prime order. The axe rank of G, studied by Ray [10], is the minimum number of spheres in a product of spheres admitting a free linear action of G. Extending a question of Ray, we conjecture that the two ranks are equal. We prove the conjecture in some special cases, including that where the axe rank is 1 or 2. We also discuss some relations between our conjecture and some questions about Bieberbach groups and free actions on tori. 2000 Mathematics Subject Classification. Primary: 20D15; Secondary: 20J05, 57S25.

On Placement of Prime Order Elements in a Group

We characterize a class of T0-groups related to the infinite Burnside groups of odd period.

P-Groups with Černikov Centralizers of Non-Identity Elements of Prime Order

Let G be a p-group, a its element of prime order p, and CG(a) a Cernikov group. We prove that either G is a Cernikov group, or G possesses a non-locally finite section w.r.t. a Cernikov subgroup in which a maximal locally finite subgroup containing an image of a is unique. Moreover, it is shown that the set of groups which satisfy the first part of the alternative is countable, while the set of groups which comply with the second is of the power of the continuum for every odd p.

Unipotent elements, tilting modules, and saturation

The results in this paper were motivated by work of Serre [19] where semisimplicity results for representations of arbitrary groups in positive characteristic were established. A key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1-dimensional unipotent subgroup. One of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups. As a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type A1 and that the Lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup. The tilting decompositions are applied to show that the centralizers of the unipotent element, a corresponding 1-dimensional unipotent group, and its Lie algebra all coincide and are closely related to the centralizer of the A1 subgroup. We also establish a convenient factorization of the centralizer of the unipotent element. Results for finite groups of Lie type are also obtained.

Zeros of characters of finite groups

There are some variations of theorems A and B which are simply not true. For instance, if χ in Irr(G) has degree divisible by p, then there does not necessarily exist a p-element on which χ vanishes. It is enough to consider L2(11) with any character of degree 10 and p = 2. It is also not true that if χ vanishes on some element x, then χ has to vanish on some p-part of x. For instance, if G is M11, then χ has an irreducible character of degree 11, vanishing on an element of order 6 and which is nonzero on 2and 3-elements. Thirdly, it is not true that a nonlinear character has to vanish on some element of prime order, as shown by any quaternion group. Interestingly enough, this seems to be the case for simple groups (and we do prove this for the groups of Lie type and the sporadic groups).

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.

The existence off-local subgroups in a group

An infinite subgroup with nontrivial finite radical is called f-local. We study the structure of subgroups of an infinite fan (set), which contain an element of prime order not embeddable in an f-local subgroup. It is proved that f-local and infinite Abelian subgroups exist in groups under certain finiteness conditions.

On elements of prime order in a group

On elements of prime order in a group

On extensions of the Baer-Suzuki Theorem

We find a necessary and sufficient condition for an element of prime order in a finite group to be in a normalp-subgroup. This generalizes the Baer-Suzuki Theorem. Our proof depends on a result about elements of prime order contained in a unique maximal subgroup containing a result of Wielandt. We discuss various consequences, linear and algebraic group versions of the result.

Commutator conditions and splitting automorphisms for stable groups

We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.

Corrigendum and Addendum to "Classification of Finite Groups with all Elements of Prime Order"

It is shown that the groups in question are either p-groups of exponent p or Frobenius groups of particular type, or they are isomorphic to the simple group A5 ; the misprints and mistakes of a previous paper of the second author are corrected.

Corrigendum and addendum to: ‘‘Classification of finite groups with all elements of prime order” [Proc.\ Amer.\ Math.\ Soc.\ {\bf 106} (1989), no.\ 3, 625–629; MR0969518 (89k:20038)] by Deaconescu

Corrigendum and addendum to: ‘‘Classification of finite groups with all elements of prime order” [Proc.\ Amer.\ Math.\ Soc.\ {\bf 106} (1989), no.\ 3, 625–629; MR0969518 (89k:20038)] by Deaconescu

LIE RINGS AND LIE GROUPS ADMITTING AN ALMOST REGULAR AUTOMORPHISM OF PRIME ORDER

It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.

The rationality of elements of prime order in compact connected simple Lie groups

We derive polynomial formulas for the number of conjugacy classes of elements of prime order (not dividing the order of the Weyl group) of a compact connected simple Lie group whose character values generate a given extension field of Q . This leads to a simple criterion for the existence of such an element.