subgrup

Subgrup, also called a subgroup in English, is a fundamental concept in group theory. A subgrup H of a group G is a subset H ⊆ G that is itself a group under the same binary operation as G. It is common to write H ≤ G to indicate that H is a subgrup of G.

To be a subgrup, H must satisfy three conditions: it is nonempty, it is closed under the group operation, and it contains inverses for all its elements. Because associativity holds in G, it also holds in H, so H is a group in its own right.

Examples help illustrate the idea. In the additive group (Z, +), every subgrup has the form nZ = {nk | k ∈ Z} for some nonnegative integer n; examples include 2Z, 3Z, and {0}. The group of nonzero real numbers under multiplication, R*, has several subgrup, such as R+ = {x > 0} and the two-element subgrup {1, -1}. The symmetric group S_n contains the alternating group A_n as a subgrup of index 2.

Normal subgrup and quotient groups are important refinements. If gHg^-1 = H for all g ∈ G, H is a normal subgrup of G, denoted H ⊴ G. Normal subgrups allow the construction of quotient groups G/H. In abelian groups, every subgrup is normal.

Other properties include that the intersection of any collection of subgrup is a subgrup. If G is finite, the order of a subgrup H divides the order of G (Lagrange’s theorem), and the index [G : H] equals the number of cosets of H in G.