Subgroup

Definitions

1. A group within a larger group; a group whose members are some, but not all, of the members of a larger group.
2. A subset H of a group G that is itself a group and has the same binary operation as G.

Equivalents

Examples

“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, 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, […] .”

“The most important application of dysprosium and terbium, which belong⟳ to a subgroup known as the heavy rare earths, is in devices called neodymium boron magnets, or neo magnets for short.”

“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.”

“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.”

“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.”

CEFR level