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## Some results from group theory

Definition 4   Let (G,.) be a (multiplicative) group with neutral element e. A nonempty subset H G is a subgroup of G if the following three statements hold
• e H,
• for every x, y H we have x y H,
• for every x H the inverse x-1 of x belongs to H.

Theorem 3   For every subgroup H of the additive abelian group (, +) there exists an element a such that H is the set of the multiples of a, that is H = a.

Theorem 4   Let G be a multiplicative group with neutral element e. Let x G an element and gr(x) the subgroup of H consisting of all powers of x (including e = x0 and x-1 the inverse of x). Let (x) be the order of gr(x), that is the cardinality of gr(x). Then two cases arise
• either gr(x) is infinite and then the powers of x are pairwise different and thus H is isomorphic to .
• or gr(x) is finite and we have the following properties
1. (x) is the smallest integer n such that xn = e,
2. xm = xm' iff m m'mod (x),
3. H is isomorphic to /n where n = (x),
4. H = {e, x, x2,..., xn-1} where n = (x).

Theorem 5 (Lagrange)   For every subgroup H of the finite group G, the order (that is the cardinality) of H divides that of G.

Next: Primitive n-th roots of unity Up: Computing primitive n-th roots of Previous: Computing primitive n-th roots of
Marc Moreno Maza
2004-04-27