Free groups
... A word w is called reduced if it contains no subword of the type ss −1 or s −1 s, for all s ∈ S. Definition A group G is called a free group if there exists a generating set S in G such that every non-empty reduced group word in S defines a non-trivial element of G . If this is the case, then one sa ...
... A word w is called reduced if it contains no subword of the type ss −1 or s −1 s, for all s ∈ S. Definition A group G is called a free group if there exists a generating set S in G such that every non-empty reduced group word in S defines a non-trivial element of G . If this is the case, then one sa ...
Building new topological spaces through canonical maps
... but it’s hard to picture what this looks like for n ≥ 3. (For n = 3, you can picture the 3-ball D3 sitting in R3 , but then how would you gather up the boundary sphere and glue it all to a single point? The resulting 3-sphere S 3 lives naturally in four dimensions, which isn’t so easy to imagine!) A ...
... but it’s hard to picture what this looks like for n ≥ 3. (For n = 3, you can picture the 3-ball D3 sitting in R3 , but then how would you gather up the boundary sphere and glue it all to a single point? The resulting 3-sphere S 3 lives naturally in four dimensions, which isn’t so easy to imagine!) A ...
Fourier analysis on abelian groups
... We now come to the first really non-trivial result about these spaces. There are two equivalent forms of this result. The first is phrased in terms of maximal translationinvariant subspaces: Proposition 1.3 (Gelfand-Mazur theorem, special case). All maximal translationinvariant subspaces are hyperpl ...
... We now come to the first really non-trivial result about these spaces. There are two equivalent forms of this result. The first is phrased in terms of maximal translationinvariant subspaces: Proposition 1.3 (Gelfand-Mazur theorem, special case). All maximal translationinvariant subspaces are hyperpl ...
