Definition of "subgroup"

Based on U.S. Bureau of the Census (1992c), other metropolitan areas that might be suitable for oversampling specific racial/ethnic subgroups include Miami (18% Cuban), New York City (7% Puerto Rican), Los Angeles (26% Mexican), and Honolulu (23% Japanese). Three techniques might be used to increase the yield of rare subgroup members within metropolitan areas where they are concentrated: 1) oversampling of areal segments containing high percentages of the subgroup, […] .

1998, Robert A. Johnson, Prevalence of Substance Use Among Racial and Ethnic Subgroups in the United States, 1991-1993, Department of Health and Human Services, page B-11

Much of the information about a group can be gleaned from a study of its subgroups. For these reasons it is important to study the subgroup structure of the almost simple groups, and in particular their maximal subgroups.

1990, Peter B. Kleidman, Martin W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press, page 1

A subgroup H of an algebraic group G is called algebraic if H is an algebraic subvariety of G. Algebraic subgroups defined over k (as algebraic subvarieties) are called k-subgroups. An algebraic subgroup of an algebraic group is called k-closed or closed over k (resp. k-defined or defined over k) if it is k-closed (resp. k-defined) as an algebraic subvariety.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, page 13

This is applied in Chapter 9 to prove the first congruence subgroup theorem, which asserts that g.z = z for all z in the center of the Drinfel'd double D(H) and all g in the principal congruence subgroup.

2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 3