Elementary Abelian Research Articles

Affine flag graphs and classification of a family of symmetric graphs with complete quotients

A graph Γ is G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. If V(Γ) admits a nontrivial G-invariant partition B such that for blocks B,C∈B adjacent in the quotient graph ΓB of Γ relative to B, exactly one vertex of B has no neighbour in C, then Γ is called an almost multicover of ΓB. In this case an incidence structure with point set B arises naturally, and it is a (G,2)-point-transitive and G-block-transitive 2-design if in addition ΓB is a complete graph. In this paper we classify all G-symmetric graphs Γ such that (i) B has block size |B|≥3; (ii) ΓB is complete and almost multi-covered by Γ; (iii) the incidence structure involved is a linear space; and (iv) G contains a regular normal subgroup which is elementary abelian. This classification together with earlier results in Gardiner and Praeger (2018), Giulietti et al. (2013) and Fang et al. (2016) completes the classification of symmetric graphs satisfying (i) and (ii).

Elementary abelian $$\varvec{p}$$ p -groups are the only finite groups with the Borsuk–Ulam property

It is well known that the Borsuk–Ulam theorem holds for elementary abelian p-groups \(C_p{}^k\). When the Borsuk–Ulam theorem holds for a finite group G, we say that G has the Borsuk–Ulam property or G is a BU-group. In this paper, we show that a non-abelian p-group of exponent p is not a BU-group, which leads to a complete classification of finite BU-groups, namely finite BU-groups are only elementary abelian p-groups.

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

We consider latin square graphs $$\varGamma = \text {LSG}(H)$$ of the Cayley table of a given finite group H. We characterize all pairs $$(\varGamma ,G)$$ , where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on $$\varGamma $$ and all nontrivial G-normal quotient graphs of $$\varGamma $$ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $$k = 1$$ or $$k = 2$$ .

On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic

We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups. This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are distinct primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$, $a_i\le 5$, are isomorphic if and only if they are isomorphic by a group automorphism of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$. That is, that such groups are CI-groups with respect to digraphs.

Fusion systems containing pearls

An F-essential subgroup is called a pearl if it is either elementary abelian of order p2 or non-abelian of order p3. In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of p-groups containing pearls and we classify the saturated fusion systems on p-groups containing pearls and having sectional rank at most 4.

Finite p-groups in which each absolute central automorphism is elementary abelian

In this paper we give necessary and sufficient conditions on a p-group G for each absolute central automorphism of G to be elementary abelian. Also we classify the absolute centre of extraspecial p-groups and show that the absolute central automorphisms of these groups are elementary abelian.

Pseudo-free families of finite computational elementary abelian p-groups

AbstractWe initiate the study of (weakly) pseudo-free families of computational elementary abelian

On the Number of Distinct Fuzzy Subgroups for Some Elementary Abelian Groups and Quaternion Groups

In this paper we compute the number of distinct fuzzy subgroups of elementary abelian groups of the form Z Z p p × and quaternion groups of the form Q4 p with respect to an equivalence relation ≈ existing in literature. The group algorithm programming (GAP) was applied in our computation.

Pentavalent symmetric graphs of order twice a prime power

A connected symmetric graph of prime valency is basic if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let p be a prime and n a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2pn, and it is shown that a connected pentavalent symmetric graph of order 2pn is basic if and only if it is either a graph of order 6, 16, 250, or a graph of three infinite families of Cayley graphs on generalized dihedral groups—one family has order 2p with p=5 or 5∣(p−1), one family has order 2p2 with 5∣(p±1), and the other family has order 2p4. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2pn have been done.It is shown that basic graphs of connected pentavalent symmetric graphs of order 2pn are symmetric elementary abelian covers of the dipole Dip5, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric Zpn-covers of the dipole Dip5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2p2 are classified.

Real groups and Sylow 2-subgroups

If G is a finite real group and P∈Syl2(G), then P/P′ is elementary abelian. This confirms a conjecture of Roderick Gow. In fact, we prove a much stronger result that implies Gow's conjecture.

Bordism of elementary abelian groups via inessential Brown–Peterson homology

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $\Omega_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $\Omega^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{Z}/p)^n$. We apply our results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.

On a class of finite capable p-groups

A group G is called capable if there is a group H such that \({G \cong H/Z(H)}\) is isomorphic to the group of inner automorphisms of H. We consider the situation that G is a finite capable p-group for some prime p. Suppose G has rank \({d(G) \ge 2}\) and Frattini class\({c \ge 1}\), which by definition is the length of a shortest central series of G with all factors being elementary abelian. There is up to isomorphism a unique largest p-group \({G_d^c}\) with rank d and Frattini class c, and G is an epimorphic image of \({G_d^c}\). We prove that this \({G_d^c}\) is capable; more precisely, we have \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\).

Rings associated to coverings of finite $p$-groups

In general the endomorphisms of a nonabelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group, which are endomorphisms when restricted to the elements of a cover of the group by abelian subgroups. We give an algorithm that allows us to determine the elements of the ring of functions of a finite $p$-group that arises in this manner when the elements of the cover are required to be either cyclic or elementary abelian of rank $2$. This enables us to determine the actual structure of such a ring as a subdirect product. A key part of the argument is the construction of a graph whose vertices are the subgroups of order $p$ and whose edges are determined by the covering.

On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian

We show that if all nonnormal subgroups of a non-Dedekindian p-group G are elementary abelian, then |G′| = p, unless G = D16 is dihedral of order 16. We also describe the p-groups of exponent > p > 2 all of whose nonnormal subgroups have exponent p. We also present a new proof of Passman’s theorem [Ber1, Theorem 1.25] classifying the non-Dedekindian p-groups all of whose nonnormal subgroups have order p. A number of related problems is stated.

The Elementary Abelian Conductor

The paper is devoted to ramification theory for a class of complete discrete valuation fields, which includes 2-dimensional local fields of prime characteristic p. It is proved that any finite extension of such a field can be modified into an extension with zero ramification depth by means of an infinite elementary Abelian base change.

On the split structure of lifted groups

Let wp: tildeX -> X be a regular covering projection of connected graphs with the group of covering transformations CT_wp being abelian. Assuming that a group of automorphisms G CT_wp -> tildeG -> G -> id splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither tildeG nor the action G -> Aut CT_wp nor a 2-cocycle GxG -> CT_wp, are given. Explicitly constructing the cover tildeX together with CT_wp and tildeG as permutation groups on tildeX is time and space consuming whenever CT_wp is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever CT_wp is elementary abelian.

Arc-transitive regular cyclic covers of the complete bipartite graph $$\mathsf{K}_{p,p}$$ K p , p

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arc-transitive regular cyclic covers of the complete bipartite graph $$\mathsf{K}_{p,p}$$Kp,p for each odd prime $$p$$p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime $$p$$p. The regular elementary abelian covers of $$\mathsf{K}_{p,p}$$Kp,p are considered in a sequel.

Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.

Derived Length of Finite p -Groups Factorable by Normal Elementary Abelian Subgroups

Derived Length of Finite p -Groups Factorable by Normal Elementary Abelian Subgroups

Constructing Pandiagonal Latin Squares from Linear Cellular Automaton on Elementary Abelian Groups

AbstractThe constructed pandiagonal Latin squares by Hedayat's method are cyclic. During the last decades several authors described methods for constructing pandiagonal Latin squares that are semi‐cyclic. In this article, we have applied linear cellular automaton on elements of permutation elementary abelian p‐groups, which are ordered by the lexicographic ordering, and we proposed an algorithm for constructing noncyclic pandiagonal Latin squares of order , for prime .