Subgroup

Every subgroup of an abelian group is abelian.

The group has only one subgroup.

The subgroup of people who had loss of smell developed more severe illness.

But when the test results are broken down by subgroup, the differences are dramatic.

In some of the experiments, stopping the consumption of wheat had a positive influence on a specific subgroup of people with schizophrenia.

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.