The classes and Fischer–Clifford matrices of extensions $$p^{1+2n}{:}G$$ and their factor groups $$p^{2n}{:}G$$

The symplectic group $ Sp_{8}(2) $ has a unique absolutely irreducible module $ 2^{16} $ of dimension 16 over $GF(2)$. Hence a split extension group $\overline{G}$ of the form $2^{16}{:}Sp_{8}(2)$ exists. The structure of the group $\overline{G}$ makes it most suitable for a standard application of the Fischer-Clifford matrices technique to compute its ordinary character table. Since this is a very large group it will not be possible to compute its character table in GAP or MAGMA with an average computer device. An existing GAP routine which computes candidates for the Fischer-Clifford matrices of an extension group, such as $\overline{G}$, also fails. This makes $\overline{G}$ a very interesting group for an application of the Fischer-Clifford matrices technique. In this paper, the authors use mostly brute force and some GAP subroutines to construct the Fischer-Clifford matrices and ordinary character table of $\overline{G}$.

In the ordinary character table of a finite group G, the values of the real valued irreducible characters on the real conjugacy classes form a sub-table which is square by Brauer's permutation lemma. We call this table the real part of the character table of G. Unlike the ordinary character table, viewed as a square matrix the real part of the character table is often singular. We present some results linking nonsingularity of this table to other properties of G.

Uniquely separable extensions

Let \(\mathcal{X}\subset\mathbb{P}_K^d\) be Drinfeld’s upper half space over a finite extension K of ℚ p . We construct for every GL d+1-equivariant vector bundle \(\mathcal{F}\) on ℙ d K , a GL d+1(K)-equivariant filtration by closed subspaces on the K-Frechet \(H^0(\mathcal{X},\mathcal{F})\). This gives rise by duality to a filtration by locally analytic GL d+1(K)-representations on the strong dual \(H^0(\mathcal{X},\mathcal{F})^{\prime}\). The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose G is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of G has finite extensions which are not residually finite. More precisely, we investigate the group H¯2(Z/n)=limΓ→H2(Γ,Z/n),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2(\\mathbb {Z}/n) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle \\varGamma }\\limits ^{\\textstyle \\rightarrow }} \\end{array} H^2(\\varGamma ,\\mathbb {Z}/n), \\end{aligned}$$\\end{document}where varGamma runs through the arithmetic subgroups of G. Elements of {bar{H}}^2(mathbb {Z}/n) correspond to (equivalence classes of) central extensions of arithmetic groups by mathbb {Z}/n; non-zero elements of {bar{H}}^2(mathbb {Z}/n) correspond to extensions which are not residually finite. We prove that {bar{H}}^2(mathbb {Z}/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion {widehat{G(mathbb {Q})}} of G(mathbb {Q}). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group H¯2(Zl)=limt←H¯2(Z/lt).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2({\\mathbb {Z}_l}) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle t}\\limits ^{\\textstyle \\leftarrow }} \\end{array} {\\bar{H}}^2(\\mathbb {Z}/l^t). \\end{aligned}$$\\end{document}We show that {bar{H}}^2({mathbb {Z}_l})^{widehat{G(mathbb {Q})}} is isomorphic to {mathbb {Z}_l}^c for some positive integer c. When G(mathbb {R}) has no simple components of complex type, we prove that c=b+m, where b is the number of simple components of G(mathbb {R}) and m is the dimension of the centre of a maximal compact subgroup of G(mathbb {R}). In all other cases, we prove upper and lower bounds on c; our lower bound (which we believe is the correct number) is b+m.

The periodic segmentation for the groups are calculated in this work from the ordinary character table and the character table of rational representations for each group.

The problem of finding the cyclic decomposition (c.d.) for the groups ), where prime upper than 9 is determined in this work. Also, we compute the Artin characters (A.ch.) and Artin indicator (A.ind.) for the same groups, we obtain that after computing the conjugacy classes, cyclic subgroups, the ordinary character table (o.ch.ta.) and the rational valued character table for each group.

The ordinary character table and the character table (cha.ta.) of rational representations (ra.repr.) for projective special linear groups (2,41) and (2,43) find in this work to find the cyclic partition for each group

This paper study compute the circularity segmentation for the projective special linear groups 𝒫𝒮𝓛(2,23) and 𝒫𝒮𝓛(2,29) from the ordinary character table and the character table of rational representations for each groups.

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT (for finite degree extensions) from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below). The derivation of the FTGT from (a) and (b) takes about four lines, but I haven’t been able to find these four lines in the literature, and all the proofs of the FTGT I have seen so far are much more complicated. So, if you find either a mistake in these four lines, or a trace of them the literature, please let me know. The argument is essentially taken from Chapter II of Emil Artin’s Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin’s proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin’s argument) was written with Keith Conrad’s help.

These conjectures should be seen as a stronger version of the integral isomorphism problem, i.e. the question whether 7ZG ~ 2gH implies G ~ H. Obviously (ZC) implies (ZCAut). If the isomorphism problem has a positive answer for G, then also the converse is true. Note that the isomorphism problem has a positive answer for all finite simple groups [12]. Thus, if G is finite simple, only the validity of (ZCAut) has to be shown in order to prove (ZC) for G. Using the ordinary character table there is an equivalent statement to (ZCAut): (ZCAut) holds for G if, and only if, for every a ~ Aut,(7lG) there exists z ~ Aut (G) which acts in the same way on the ordinary character table as ~ does. These two conjectures of Zassenhaus led to new considerations in recent years. One of the first results in this context is due to Peterson who proved that the symmetric groups S, are elementary represented which means that for S. (ZCAut) is valid [15]. It was probably this theorem that led to the formulation of the two conjectures. In general, the conjecture (ZC) is not valid, as shown by Roggenkamp and Scott who constructed a metabelian counterexample [18, IX w i]. Nevertheless, they were able to prove that (ZC) holds for nilpotent groups and for groups whose generalized Fitting group is a p-group [16, 17]. However, there is very limited knowledge with respect to simple non-abelian groups. The main result of this paper is the proof of the following two theorems.

This paper begins with an introduction to β-Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and β-Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split β-Frobenius and separable β-Frobenius extensions are proven. Sugano"s theorem is extended to β-Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable β-Frobenius extensions which are not Frobenius in the ordinary sense.

Let $Aut_kK$ be the automorphism scheme of a finite purely inseparable field extension $k\to K$ (in fact, we consider a wide class of finite ring extensions). Let $A_M$ be the splitting algebra of K (see 2.8) and $K_M={A_M}_{\text red} $ the splitting field of K. It is proved that $Aut_kK$ is integral if and only if $A_M=K_M$ . This may be formulated as a condition on the degree of $K_M$ and generalizes a result of Chase. Surprisingly, $A_M$ may be defined intrinsically, since $\mathrm{Spec} A_M$ is the scheme parametrizing the maximal smooth subgroups of $Aut_kK$ . It is also proved that the desingularization of $Aut_kK$ is the universal maximal smooth subgroup of $Aut_kK$ and coincides with the blowing up along a closed subscheme canonically defined from the action of $Aut_kK$ on $A_M$ .

If G is a group we will write Aut G for the group of all automorphisms of G and Inn G for the normal subgroup of all inner automorphisms of G. Many authors have studied the relationship between the structure of G and that of Aut G, in particular when the latter is finite. This paper is a further contribution to this study. The first results on groups whose automorphism groups are finite were published by Baer in a paper [2] in which he proved that a torsion group has finite automorphism group only if it is finite. Baer also proved that a group with only a finite number of endomorphisms is finite. In 1962 Alperin [1] characterized finitely generated groups with finitely many automorphisms as finite central extensions of cyclic groups. Nagrebeckil [9] discovered in 1972 the important result that in any group with finitely many automorphisms the elements of finite order form a finite subgroup. This of course generalizes Baer's original result. Robinson El0] has given another proof of Nagrebeckfi's Theorem as well as obtaining information on the primes dividing the order of the maximal torsion subgroup. He also characterized the center of a group whose automorphism group is finite and gave a general method for constructing examples. On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be several authors that torsion-free abelian groups with only one non-trivial automorphism the involution x F--~x~ are relatively common (de Groot [5], Fuchs [4], Corner [3]). However, Hallett and Hirsch have adopted a different approach, asking which finite groups can occur as the automorphism groups of torsion-free abelian groups. They have established the following definitive result [-7, 8]: