High Quality Content by WIKIPEDIA articles! In mathematics, in the
realm of group theory, a countable group is said to be SQ universal
if every countable group can be embedded in one of its quotient
groups. SQ-universality can be thought of as a measure of largeness
or complexity of a group. Many classic results of combinatorial
group theory, going back to 1949, are now interpreted as saying
that a particular group or class of groups is (are) SQ-universal.
However the first explicit use of the term seems to be in an
address given by Peter Neumann to the The London Algebra Colloquium
entitled "SQ-universal groups" on 23 May 1968. In 1949 Graham
Higman, Bernhard Neumann and Hanna Neumann proved that every
countable group can be embedded in a two generator group. Using the
contemporary language of SQ-universality, this result says that
F_2, the free group (non-abelian) on two generators, is
SQ-universal. This is the first known example of an SQ-universal
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