Alcsoportokra

Alcsoportokra is the Hungarian dative plural form of alcsoport, used in sentences that refer to subgroups in Hungarian. In mathematical contexts, alcsoportok means subgroups of a group, and alcsoportokra often appears with verbs that indicate division or relation to subgroups, for example in sentences like “a csoport alcsoportokra oszlik” (the group divides into subgroups).

Definition

- An alcsoport (subgroup) H of a group G is a nonempty subset of G that is closed under the group operation and contains the inverses of its elements. In additive groups this means H is closed under addition and contains the zero element.

Key properties

- Closure: if a and b are in H, then ab (or a + b) is in H.

- Identity and inverses: the identity element of G is in H, and every element of H has an inverse in H.

- Generated subgroups: the subgroup generated by a set S is the smallest alcsoport containing S.

Examples

- In the group of integers under addition, 2Z (the even integers) is an alcsoport of Z.

- In the multiplicative group of nonzero real numbers, the set {1, −1} is an alcsoport.

Related concepts

- Normal subgroups, quotient groups, and homomorphisms are central to the study of alcsoportokra in Hungarian texts, as in English-language group theory.

- The order of a finite alcsoport divides the order of the ambient group (Lagrange’s theorem), and subgroups organize the group's structure via a subgroup lattice.

Usage note

- In Hungarian grammar, alcsoportokra appears when the sentence requires the dative plural form, often conveying the meaning “to the subgroups” or “for the subgroups.” See also alcsoportok for the nominative plural.