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The intersection of subgroups A and B of G is again a subgroup of G. [5] For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: smoking status defining two subgroups: smokers and non-smokers.
Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G {\displaystyle G} , which means that they can be used to internally classify those homomorphisms.
Verbal subgroups are the only fully characteristic subgroups of a free group and therefore represent the generic example of fully characteristic subgroups, (Magnus, Karrass & Solitar 2004, p. 75). Another example is the verbal subgroup for { x − 1 y − 1 x y } {\displaystyle \{x^{-1}y^{-1}xy\}} , which is the derived subgroup .
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the ...
Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots.
In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulikov (for p-groups) and by László Fuchs (in general) in an attempt to formulate classification theory of infinite abelian groups that goes beyond the Prüfer theorems.
In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group.. Given a group , the center of , denoted as (), is defined as the set of those elements of the group which commute with every element of the group.