Search results
Results from the WOW.Com Content Network
This statement is known by various names such as characterization by subgroups. [5] [6] [7] (See also cyclic group for some characterization.) There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. However, the Klein group has more than one subgroup of order 2 ...
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...
The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.)
Let V be the non-cyclic subgroup of A 4 called the Klein four-group. V = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. Let K = H ⋂ V. Since both H and V are subgroups of A 4, K is also a subgroup of A 4. From Lagrange's theorem, the order of K must divide both 6 and 4, the orders of H and V respectively. The only two positive integers that divide ...
Cauchy's theorem is generalized by Sylow's first theorem, which implies that if p n is the maximal power of p dividing the order of G, then G has a subgroup of order p n (and using the fact that a p-group is solvable, one can show that G has subgroups of order p r for any r less than or equal to n).
Proof. Let Ω be the set of all Sylow p-subgroups of G and let G act on Ω by conjugation. Let P ∈ Ω be a Sylow p-subgroup. By Theorem 2, the orbit of P has size n p, so by the orbit-stabilizer theorem n p = [G : G P]. For this group action, the stabilizer G P is given by {g ∈ G | gPg −1 = P} = N G (P), the normalizer of P in G.
The ping-pong lemma is also used for studying Schottky-type subgroups of mapping class groups of Riemann surfaces, where the set on which the mapping class group acts is the Thurston boundary of the Teichmüller space. [8] A similar argument is also utilized in the study of subgroups of the outer automorphism group of a free group. [9]