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Category Archives: exposition
GRA Workshop 1
Last week was the introductory/instructional workshop for the Isaac Newton Institute’s six-month programme on Groups, Representations and Applications. We were thrown in the deep end right at the start. The first two talks were on what were claimed to be … Continue reading
The Frattini argument
The Frattini subgroup of a finite group G can be defined in two equivalent ways: it is the intersection of all the maximal proper subgroups of G; it is the set of all non-generators of G, that is, elements which … Continue reading
Sylow’s Theorem, from the book
The most important theorems of elementary group theory are those of Lagrange and Sylow. I want to describe here what I consider the most beautiful proof of the first part of Sylow’s Theorem, actually based on Sylow’s original proof. But … Continue reading
Something I didn’t know
I didn’t know this, though probably I should have. Maybe you didn’t know it either. We work in a semigroup, a system with an operation (called multiplication) satisfying the associative law. A generalised inverse of an element A is an … Continue reading
Posted in exposition
Tagged associative law, commute, generalised inverse, matrices, polynomial
4 Comments
The Wall conjecture extended
Tim Wall boldly conjectured in 1961 that the number of maximal subgroups of a finite group G is at most |G|−1. (This would be best possible, since the the elementary abelian 2-group attains this bound.) He proved that the conjecture … Continue reading
A challenge
Let a(n) be defined in the following manner. Write n as an ordered sum of positive integers in all possible ways. (The number of such expressions is 2n−1; showing this is a pleasant exercise, but it is not relevant for … Continue reading
G2D2, 4: the second week
The lecturers of the first two minicourses now gave way for the second team. Mike Boyle and Scott Schmieding told us about symbolic dynamics, in particular, subshifts and their automorphism groups. This was particularly valuable to me: I had dove … Continue reading
