Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 1 , ⋯ , m } and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.

A generalization of Sims’ conjecture for finite primitive groups and two point stabilizers in primitive groups

In this paper, we propose a refinement of Sims’ conjecture concerning the cardinality of the point stabilizers in finite primitive groups, and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group 𝐺 on Ω and two distinct points α , β ∈ Ω \alpha,\beta\in\Omega with G α ⁢ β ⊴ G α G_{\alpha\beta}\unlhd G_{\alpha} and G α ⁢ β ≠ 1 G_{\alpha\beta}\neq 1 , where G α G_{\alpha} is the stabilizer of 𝛼 in 𝐺 and G α ⁢ β G_{\alpha\beta} is the stabilizer of 𝛼 and 𝛽 in 𝐺. In particular, this example gives an answer to a question raised independently by Cameron and by Fomin in the Kourovka Notebook.

On the involution fixity of simple groups

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$ , which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$ . This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

In this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

A note on extremely primitive affine groups

Let G be a finite primitive permutation group on a set $$\Omega $$ Ω with non-trivial point stabilizer $$G_{\alpha }$$ G α . We say that G is extremely primitive if $$G_{\alpha }$$ G α acts primitively on each of its orbits in $$\Omega {\setminus } \{\alpha \}$$ Ω \ { α } . In earlier work, Mann, Praeger, and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall’s conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.

TRIBAL ORGAN DRAWING ART - TATTOO

English : The wild race or tribe is known by the name of primitive, tribal, Vanvasi Girijan Scheduled Tribe, these are called primitive or tribal because they are considered to be the oldest inhabitants of India. These people used to live here before the arrival of the Dravidians in India. What is primitive culture after all? Efforts are being made to know, understand the primitive culture all over the world, the more we have learned about the primitive groups, the more are left to learn more. Regardless of how civilized and modern we have become today, but it is true that even today the number of tribals in our society is two-thirds. If we look at the history of the primitive groups adopting their own personal culture and way of life, then the tribal society, unlike the society, lived life with its primitive energy and power in every situation. Hindi : वन्य जाति या जनजाति को आदिम, आदिवासी, वनवासी गिरिजन अनुसूचित जनजाति के नामों से जाना जाता है, इन्हें हम आदिम या आदिवासी इसलिए कहते हैं, क्योंकि ये भारत के सबसे प्राचीनतम निवासी माने जाते हैं । भारत में द्रविडों के आगमन से पूर्व यहाँ ये ही लोग निवास करते थे। आदिम संस्कृति आखिर है क्या ? आदिम संस्कृति को जानने, समझने का प्रयत्न संपूर्ण विश्व में हो रहा है, जितने पहलु हमने आदिम समूहों के बारे में जान लिए हैं, उतने ही और भी जानने के लिए शेष बचे हैं । भले ही हम आज कितने ही सभ्य और आधुनिक कहलाने लगे हैं, लेकिन यह सत्य है कि, हमारे समाज में आज भी आदिवासियों की संख्या दो तिहाई है1 । अपनी निजी संस्कृति एवं जीवन पद्धति को अपनाये आदिम समूहों के इतिहास को देखें तो लोग समाज के समानान्तर आदिवासी समाज भी हर परिस्थिति में अपनी आदिम ऊर्जा और शक्ति के साथ जीवन जीता रहा ।