## João Xarez

## Generalising the Galois theorem

The correspondence between Galois extensions and field automorphisms, in classical Galois theory, can be seen as a special case of a much more general theory. We will try to give a brief explanation of why this is so, illustrating with some examples and results of our own research.

Maria Elisa Fernandes

## Highest rank of a polytope for A_n

The existence of a regular polytope with a given automorphism group G can be translated into a group-theoretic condition on a generating set of involutions for G. For G being the symmetric group S_n, the maximum rank of such a polytope is n-1, with equality only for the regular simplex.

We prove that the highest rank of a string C-group constructed from an alternating group A_n is 0 if n=3, 4, 6, 7, 8; 3 if n=5; 4 if n=9; 5 if n=10; 6 if n=11; and the largest integer not greater than (n-1)/2 if n is greater or equal than 12.

This is a joint work with Peter Cameron, Dimitri Leemans and Mark Mixer.

We prove that the highest rank of a string C-group constructed from an alternating group A_n is 0 if n=3, 4, 6, 7, 8; 3 if n=5; 4 if n=9; 5 if n=10; 6 if n=11; and the largest integer not greater than (n-1)/2 if n is greater or equal than 12.

This is a joint work with Peter Cameron, Dimitri Leemans and Mark Mixer.