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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 $ \subseteq$ G is a subgroup of G if the following three statements hold

Theorem 3   For every subgroup H of the additive abelian group ($ \mbox{${\mathbb Z}$}$, +) there exists an element a $ \in$ $ \mbox{${\mathbb Z}$}$ such that H is the set of the multiples of a, that is H = a$ \mbox{${\mathbb Z}$}$.

Theorem 4   Let G be a multiplicative group with neutral element e. Let x $ \in$ 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 $ \Theta$(x) be the order of gr(x), that is the cardinality of gr(x). Then two cases arise

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 up previous
Next: Primitive n-th roots of unity Up: Computing primitive n-th roots of Previous: Computing primitive n-th roots of
Marc Moreno Maza
2003-06-06