1.
Finite group
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In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, with a finite group, the set is finite. As a consequence, the classification of finite simple groups was achieved. During the second half of the century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups. One such family of groups is the family of linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, the theory of Lie groups, which may be viewed as dealing with continuous symmetry, is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space, the properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry. Since there are n. possible permutations of a set of n symbols, a cyclic group Zn is a group all of whose elements are powers of a particular element a where an = a0 = e, the identity. A typical realization of this group is as the nth roots of unity. Sending a to a root of unity gives an isomorphism between the two. This can be done with any finite cyclic group, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. They are named after Niels Henrik Abel, an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of an abelian group can be described directly in terms of these invariants. A group of Lie type is a closely related to the group G of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian simple groups. Special cases include the groups, the Chevalley groups, the Steinberg groups. The systematic exploration of finite groups of Lie type started with Camille Jordans theorem that the special linear group PSL is simple for q ≠2,3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL of finite simple groups, other classical groups were studied by Leonard Dickson in the beginning of 20th century

2.
Alternating group
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In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the group of degree n, or the alternating group on n letters. For n >1, the group An is the subgroup of the symmetric group Sn with index 2 and has therefore n. It is the kernel of the signature group homomorphism sgn. The group An is abelian if and only if n ≤3 and simple if, a5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has a Klein four-group V as a normal subgroup, namely the identity and the double transpositions. As in the group, the conjugacy classes in An consist of elements with the same cycle shape. Examples, The two permutations and are not conjugates in A3, although they have the same cycle shape, the permutation is not conjugate to its inverse in A8, although the two permutations have the same cycle shape, so they are conjugate in S8. An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions and this generating set is often used to prove that An is simple for n ≥5. For n =1 and 2, the group is trivial. For n =3 the automorphism group is Z2, with trivial inner automorphism group, the outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles with elements of shape 32, there are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are, A4 is isomorphic to PSL2 and the group of chiral tetrahedral symmetry. A5 is isomorphic to PSL2, PSL2, and the group of chiral icosahedral symmetry. A6 is isomorphic to PSL2 and PSp4, more obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group. A subgroup of three elements with any additional element generates the whole group, for all n ≠4, An has no nontrivial normal subgroups. Thus, An is a group for all n ≠4. A5 is the smallest non-solvable group, the group homology of the alternating groups exhibits stabilization, as in stable homotopy theory, for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology, note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions