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Category Archives: Exercise

Leo’s first theorem

I learnt the following from my son Léo: the teacher asks to compute ; that’s easy But no! The actual question is to compute times ! We must correct this! But it’s just as easy without starting from scratch: we turn the “plus” cross a quarter turn on the left-hand side: and then switch the […]

The discrete spectrum is discrete

No, this post is not an exercise in tautological reasoning: the point is that the word “discrete” is relatively overloaded. In the theory of automorphic forms, “discrete spectrum” (or “spectre discret”) is the same as “cuspidal spectrum”, and refers to those automorphic representations (of a given group over a given global field ) which are […]

More conjugation shenanigans

After I wrote my last post on the condition in a group, I had a sudden doubt concerning the case in which this arose: there we assume that we have a coset such that for all . I claimed that this implies , but really the argument I wrote just means that : for all […]

Normalizers everywhere

In working on a paper, I found myself in the amusing but unusual situation of having a group , a subgroup and an element such that This certainly can happen: the two obvious cases are when , or when is an involution that happens to be in the normalizer of . In fact the general […]

Three little things I learnt recently

In no particular order, and with no relevance whatsoever to the beginning of the year, here are three mathematical facts I learnt in recent months which might belong to the “I should have known this” category: (1) Which finite fields have the property that there is a “square root” homomorphism i.e., a group homomorphism such […]