Problem Set - Groups and Monoids

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Groups and Monoids

	Let $S$ be a set with an associative operation with identity. Show that the identity is unique. (This tells us that any commutative monoid has only one heart.)
	Let $S$ be a set with an associative operation with identity. Let $s\in S$ and assume that $s$ has an inverse in $S$. Show that the inverse of $s$ is unique. (This tell us that any element of an abelian group has only one mate.)
	Let $S$ be a set with identity. Let $s\in S$ and assume that $s$ has an inverse in $S$. Show that the inverse of the inverse of $s$ is equal to $s$. (This tells us that $-(-s)=s$.)
	Let $S$ be an abelian group. Show that if $a+c=b+c$ then $a=b$.
	Let $S$ be a ring. Show that if $s\in S$ then $s\cdot 0=0$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)