Narrow Results

Let P be a field of characteristic different from 2, let K be an associative commutative P-algebra with an identity 1 and let n be an integer, n ≥ 2. Assume that K is an algebraic extension of P having, in general, zero divisors and P is an algebraic separable extension of an infinite subfield k. The paper studies subgroups X of the group GL_n (K) such that X contains a root k-subgroup, i.e. a subgroup which is conjugate in GL_n (K) to a group of all matrices , a ∈ k.

In this paper we exhibit a way of obtaining linear representations of the braid groups B_n over ℤ[t] by studying their action on the set of isotopy classes of sets of simple closed curves on a punctured disc. The cases n=3, 4, 5 are shown to be very different from the cases n>5. We show a connection between representations of B₃ and Pascal's triangle. We also show that there is a sequence of polynomials κ_i(t), i≥0, related to polynomials P_i(t) defined by V. F. R. Jones all of whose roots give values of t for which these representations are not faithful.

Let G be a finite group and let cd(G) be the set of all irreducible complex character degrees of G. It was conjectured by Huppert in Illinois J. Math.44 (2000) that, for every non-abelian finite simple group H, if cd(G) = cd(H) then G ≅ H × A for some abelian group A. In this paper, we confirm the conjecture for the family of projective special linear groups PSL₄(q) with q ≥ 13.

Let G be a finite group and π_e(G) be the set of element orders of G. Assume that k ∈ π_e(G) and m_k(G) is the number of elements of order k in G. Set nse(G) ≔ {m_k(G) | k ∈ π_e(G)}, we call nse(G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L₂(2^a) by its order |L₂(2^a)| and the set nse(L₂(2^a)), where either 2^a - 1 or 2^a + 1 is a prime.

We prove that if $𝔽$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $\phi$ of ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ such that $R(\phi )=1$. This fact implies that ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ do not possess the $R_{\infty }$-property. However, if the transcendece degree of $𝔽$ over $\mathbb{Q}$ is finite, then ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ are known to possess the $R_{\infty }$-property [13].

In this paper, we show that for any finite field $K$, any pair of map-generators (that is when one of the generators is an involution) of $\mathrm{\text{SL}}(3,K)$ and $\mathrm{\text{PSL}}(3,K)$ has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group $\mathrm{\text{SL}}(3,K)$ or $\mathrm{\text{PSL}}(3,K)$ is reflexible, or equivalently, there are no chiral regular maps with automorphism group $\mathrm{\text{SL}}(3,K)$ or $\mathrm{\text{PSL}}(3,K)$. As remarked by Leemans and Liebeck, also $\mathrm{\text{SU}}(3,K)$ and $\mathrm{\text{PSU}}(3,K)$ are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

Answering a question of G. Yu, we show that the group EL₃(ℤ 〈 x, y 〉) is not linear over any field. We also prove that for every unitary associative ring R, the group EL_k(R) (k ≥ 3) is linear over ℂ if and only if R has a finite index ideal that is linear over ℂ.

In the paper, we study finitely generated linear groups of finite rank which have faithful irreducible primitive representations over a field of characteristic zero. We prove that if an infinite finitely generated linear group $G$ of finite rank has a faithful irreducible primitive representation over a field of characteristic zero then the $FC$-center $\mathrm{\Delta }(G)$ of $G$ is infinite.