sousgroupes

In mathematics, sousgroupes (subgroups) are a central concept in group theory. If G is a group with a binary operation, a subset H ⊆ G is called a sousgroupe if H itself forms a group under the same operation. Equivalently, the identity of G must lie in H, H must be closed under the group operation, and every element of H must have its inverse in H. When these conditions hold, H is denoted H ≤ G; if H is a proper subset, it is called a proper sousgroupe, written H < G.

Examples help clarify the idea. In the additive group of integers Z, the set of even integers 2Z is a sousgroupe; the trivial subgroup {0} and the whole group Z are also sousgroupes. In a finite group, the order (size) of any sousgroupe H divides the order of G, a result known as Lagrange’s theorem. Subgroups inherit the structure of the ambient group, and their cosets partition G.

Key properties include that the intersection of any collection of sousgroupes is a sousgroupe, while the union of two sousgroupes need not be a sousgroupe. A sousgroupe H is normal, written H ◁ G, when gH = Hg for all g in G; in that case one can form the quotient group G/H, whose elements are cosets and whose operation is defined by coset multiplication.

Generation and homomorphisms are fundamental tools. The sousgroupe generated by a subset S ⊆ G, denoted ⟨S⟩, is the smallest sousgroupe containing S. If φ: G → K is a group homomorphism and H ≤ G, then φ(H) ≤ φ(G) is a sousgroupe of K; the kernel and image are examples of subgroups arising naturally from φ.