  The sumset is one of the most basic and central objects in additive number
theory. Many of the most important problems (such as Goldbach's conjecture and
Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y
: x,y\in S\}$ of a set of integers $S$. A finite set of integers $A$ is
sum-dominated if $|A+A| > |A-A|$. Though it was believed that the percentage of
subsets of $\{0,...,n\}$ that are sum-dominated tends to zero, in 2006 Martin
and O'Bryant proved a very small positive percentage are sum-dominated if the
sets are chosen uniformly at random (through work of Zhao we know this
percentage is approximately $4.5 \cdot 10^{-4}$). While most sets are
difference-dominated in the integer case, this is not the case when we take
subsets of many finite groups. We show that if we take subsets of larger and
larger finite groups uniformly at random, then not only does the probability of
a set being sum-dominated tend to zero but the probability that $|A+A|=|A-A|$
tends to one, and hence a typical set is balanced in this case. The cause of
this marked difference in behavior is that subsets of $\{0,..., n\}$ have a
fringe, whereas finite groups do not. We end with a detailed analysis of
dihedral groups, where the results are in striking contrast to what occurs for
subsets of integers.
