alcsoportja

In Hungarian mathematical terminology, alcsoportja of a group G refers to a subgroup. A subset H of G is an alcsoportja (subgroup) if H itself forms a group under the same operation as G. This is usually written as H ≤ G.

To be an alcsoportja, H must satisfy the subgroup criterion: it is nonempty, closed under the group operation, and closed under taking inverses. Equivalently, the identity element of G lies in H, and for any a and b in H, their product a b is in H, and the inverse a^{-1} is in H. These conditions ensure H inherits the group structure from G.

Subgroups have several important properties. The intersection of any family of subgroups of G is again a subgroup. If S is a subset of G, the subgroup generated by S is the smallest alcsoportja containing S. For a single element a in G, the set of all powers of a, denoted <a>, is the cyclic alcsoportja generated by a.

Common examples illustrate the concept. The set of even integers is a subgroup of the integers under addition. The trivial set {e} and the whole group G are both subgroups of G. Subgroups can be finite or infinite and may be cyclic or more complex in structure.

Related notions include normal subgroups, which are subgroups invariant under conjugation, and quotient groups formed by dividing G by a normal subgroup. The index [G:H] measures how many cosets of H exist in G, and Lagrange’s theorem relates this index to the order of finite subgroups. Subgroup structure is fundamental to understanding the composition and properties of a group.