Let k be an algebraically closed field, G a linear algebraic group over k and φ ∈ Aut(G), the group of all algebraic group automorphisms of G. Two elements x; y of G are said to be φ-twisted conjugate if y = gxφ(g)–1 for some g ∈ G. In this paper we prove that for a connected non-solvable linear algebraic group G over k, the number of its φ-twisted conjugacy classes is infinite for every φ ∈ Aut(G).

Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R∞-property, then Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R∞-property and identify certain classes of solvable algebraic groups for which the property fails.

Let G be a group. An element x ∈ G is called real if x is conjugate to x−1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F4 and algebraic groups of type F4 defined over an arbitrary field.

We compute the number of orbit types for simply connected simple algebraic groups over algebraically closed fields as well as for compact simply connected simple Lie groups. We compute the number of orbit types for the adjoint action of these groups on their Lie algebras. We also prove that the genus number of a connected reductive algebraic group coincides with the genus number of its semisimple part.