In mathematics, a quasirandom group is a group that does not contain a large product-free^{[clarification needed]} subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake of these groups stems from their connection to graph theory: bipartite Cayley graphs over any subset of a quasirandom group are always bipartite quasirandom graphs.

The notion of quasirandom groups arises when considering subsets of groups for which no two elements in the subset have a product in the subset; such subsets are termed product-free. László Babai and Vera Sós asked about the existence of a constant ${\displaystyle c}$ for which every finite group ${\displaystyle G}$ with order ${\displaystyle n}$ has a product-free subset with size at least ${\displaystyle cn}$. A well-known result of Paul Erdős about sum-free sets of integers can be used to prove that ${\textstyle c={\frac {1}{3}}}$ suffices for abelian groups, but it turns out that such a constant does not exist for non-abelian groups.

Both non-trivial lower and upper bounds are now known for the size of the largest product-free subset of a group with order ${\displaystyle n}$. A lower bound of ${\textstyle cn^{\frac {11}{14}}}$ can be proved by taking a large subset of a union of sufficiently many cosets, and an upper bound of ${\textstyle cn^{\frac {8}{9}}}$ is given by considering the projective special linear group ${\displaystyle \operatorname {PSL} (2,p)}$ for any prime ${\displaystyle p}$. In the process of proving the upper bound, Timothy Gowers defined the notion of a quasirandom group to encapsulate the product-free condition and proved equivalences involving quasirandomness in graph theory.

Formally, it does not make sense to talk about whether or not a single group is quasirandom. The strict definition of quasirandomness will apply to sequences of groups, but first bipartite graph quasirandomness must be defined. The motivation for considering sequences of groups stems from its connections with graphons, which are defined as limits of graphs in a certain sense.

Fix a real number ${\displaystyle p\in [0,1].}$ A sequence of bipartite graphs ${\displaystyle (G_{n})}$ (here ${\displaystyle n}$ is allowed to skip integers as long as ${\displaystyle n}$ tends to infinity) with ${\displaystyle G_{n}}$ having ${\displaystyle n}$ vertices, vertex parts ${\displaystyle A_{n}}$ and ${\displaystyle B_{n}}$, and ${\displaystyle (p+o(1))|A_{n}||B_{n}|}$ edges is quasirandom if any of the following equivalent conditions hold:

It is a result of Chung–Graham–Wilson that each of the above conditions is equivalent. Such graphs are termed quasirandom because each condition asserts that the quantity being considered is approximately what one would expect if the bipartite graph was generated according to the Erdős–Rényi random graph model; that is, generated by including each possible edge between ${\displaystyle A_{n}}$ and ${\displaystyle B_{n}}$ independently with probability ${\displaystyle p.}$

Though quasirandomness can only be defined for sequences of graphs, a notion of ${\displaystyle c}$-quasirandomness can be defined for a specific graph by allowing an error tolerance in any of the above definitions of graph quasirandomness. To be specific, given any of the equivalent definitions of quasirandomness, the ${\displaystyle o(1)}$ term can be replaced by a small constant ${\displaystyle c>0}$, and any graph satisfying that particular modified condition can be termed ${\displaystyle c}$-quasirandom. It turns out that ${\displaystyle c}$-quasirandomness under any condition is equivalent to ${\displaystyle c^{k}}$-quasirandomness under any other condition for some absolute constant ${\displaystyle k\geq 1.}$

The next step for defining group quasirandomness is the Cayley graph. Bipartite Cayley graphs give a way from translating quasirandomness in the graph-theoretic setting to the group-theoretic setting.