Let k k be any field and let G G be a connected reductive algebraic k k -group. Associated to G G is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of G G (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the k k -isogeny class of G G is uniquely determined by its index and the k k -isogeny class of its anisotropic kernel G a G_a . For the cases where G G is absolutely simple, all possibilities for the index of G G have been classified in by Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62]. Let H H be a connected reductive k k -subgroup of maximal rank in G G . We introduce an invariant of the G ( k ) G(k) -conjugacy class of H H in G G called the embedding of indices of H ⊂ G H \subset G . This consists of the index of H H and the index of G G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k k -subgroups of G G , and observe that the G ( k ) G(k) -conjugacy class of H H in G G is determined by its index-conjugacy class and the G ( k ) G(k) -conjugacy class of H a H_a in G G . We show that the index-conjugacy class of H H in G G is uniquely determined by its embedding of indices. For the cases where G G is absolutely simple of exceptional type and H H is maximal connected in G G , we classify all possibilities for the embedding of indices of H ⊂ G H \subset G . Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k k has cohomological dimension 1 (resp. k = R k=\mathbb {R} , k k is p \mathfrak {p} -adic).

There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions for finite and profinite associative algebras. Let k = Fq be a finite field. Let A be a finite dimensional, associative, unital algebra over k. Let P (A) be the probability that two elements of A chosen (uniformly and independently) at random will generate A as a unital k-algebra. It is known that, if A is simple, then P (A) → 1 as |A| → ∞. We prove an analogue of this result for A an arbitrary finite associative algebra. For A simple, we find the lower bound for P (A) and we estimate the growth rate of P (A) in terms of the minimal index m(A) of any proper subalgebra of A. We also study the random generation of A by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of A. Finally, we let A be a profinite algebra over k. We show that A is positively finitely generated if and only if A has polynomial maximal subalgebra growth. Related quantitative results are also established.

Let G be a connected real algebraic group. An unrefinable chain of G is a chain of subgroups G = G 0 > G 1 > ... > G t = 1 where each G i is a maximal connected real subgroup of G i−1 . The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of G. We give a precise formula for the length of G, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups [3] and also on compact Lie groups [4]. If G is simple then we bound the depth of G above and below, and in many cases we compute the exact value. In particular, the depth of any simple G is at most 9.

We describe a straightforward construction of the pseudo-split absolutely pseudosimple groups of minimal type with irreducible root systems of type BCn; these exist only in characteristic 2. We also give a formula for the dimensions of their irreducible modules.

Let k be any field and let G be a connected reductive algebraic k-group. Associated to G is an invariant first studied by Satake [23,24] and Tits [31] that is called the index of G (a Dynkin diagram along with some additional combinatorial information). Tits [31] showed that the k-isogeny class of G is uniquely determined by its index and the k-isogeny class of its anisotropic kernel G a . For the cases where G is absolutely simple, Satake [23] and Tits [31] classified all possibilities for the index of G. Let H be a connected reductive k-subgroup of maximal rank in G. We introduce an invariant of the G(k)-conjugacy class of H in G called the embedding of indices of H ⊂ G. This consists of the index of H and the index of G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k-subgroups of G, and observe that the G(k)-conjugacy class of H in G is determined by its index-conjugacy class and the G(k)-conjugacy class of H a in G. We show that the index-conjugacy class of H in G is uniquely determined by its embedding of indices. For the cases where G is absolutely simple of exceptional type and H is maximal connected in G, we classify all possibilities for the embedding of indices of H ⊂ G. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k has cohomological dimension 1 (resp. k = R, k is p-adic).