Focal subgroup theoremIn abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described in . Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index p.
Index of a subgroupIn mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index measures the "relative sizes" of G and H.
Order (group theory)In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a.
Dihedral group of order 6In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group. This page illustrates many group concepts using this group as example. The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed.
Cyclic groupIn group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation.
