In mathematics, an alternating group is the group of permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt. For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n! / 2 elements. It is the kernel of the signature group homomorphism sgn: Sn → explained under symmetric group; the group An is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A5 is the smallest non-abelian simple group, having order 60, the smallest non-solvable group; the group A4 has a Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions, maps to A3 = C3, from the sequence V → A4 → A3 = C3. In Galois theory, this map, or rather the corresponding map S4 → S3, corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.
As in the symmetric group, any two elements of An that are conjugate by an element of An must have the same cycle shape. The converse is not true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type there are two conjugacy classes for this cycle shape. Examples: The two permutations and are not conjugates in A3, although they have the same cycle shape, are therefore conjugate in S3; 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. See Symmetric group. An is generated by 3-cycles; this generating set is used to prove that An is simple for n ≥ 5. For n > 3, except for n = 6, the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2. For n = 1 and 2, the automorphism group is trivial. For n = 3 the automorphism group is Z2, with trivial inner automorphism group and outer automorphism group Z2.
The outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, 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 projective special linear groups; these are: A4 is isomorphic to PSL2 and the symmetry group of chiral tetrahedral symmetry. A5 is isomorphic to PSL2, PSL2, the symmetry group of chiral icosahedral symmetry.. A6 is isomorphic to PSL2 and PSp4'. A8 is isomorphic to PSL4. More A3 is isomorphic to the cyclic group Z3, A0, A1, A2 are isomorphic to the trivial group. A 5 is the group of isometries of a dodecahedron in 3 space, so there is a representation A 5 → S O 3 In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element; each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians.
Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A 5 is 1+12+12+15+20=60, we obtain four distinct polyhedra; the vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of -cycles, represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by π radians, so can be represented by a vector of length π in either of two directions, thus the class of - cycles contains 15 elements. The two conjugacy classes of twelve 5-cycles in A 5 are represented by two icosahedra, of radii 2 π / 5 and 4 π / 5, respectively; the nontrivial outer automorphism in Out ≃ Z 2 interchanges these two classes and the corresponding icosahedra. A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not exist a subgroup of G with order d: the group G = A4, of order 12, has no subgroup of order 6.
A subgroup of three elements with any distinct nontrivial element generates the whole group. For all n > 4, An has no nontrivial normal subgroups. Thus, An is a simple group for all n > 4. A5 is the smallest non-solvable group; the group homology of the alternating groups exhibits
Adenophora triphylla known as Japanese lady bell, is one of the 62 species of Adenophora. It is a flowering plant in the family Campanulaceae, distributed over the Korean Peninsula and China. Adenophora triphylla is an perennial herb growing to 100 centimetres in height, it has a thickened taproot, shaped like a carrot, 7-16 × 1.5-1.8 cm in diameter. Stems are white pilose with alternately arranged leaves, it has oval round, serrated leaves growing to 10 centimetres that are white pointed, pilose. A. triphylla flowers are about 13 millimetres -22 millimetres long and have both male and female organs, each having 5 stamens and a pistil. Flowers are pollinated by insects. Seeds are yellow-brown colored and oblong compressed, 1 millimetre -1.5 millimetres. Habitat: Grassy areas in lowlands and mountains. Suitable for: Grassy places in lowland and mountain with loamy soils. Distribution: Korea, China, Russia, Vietnam. Adenophora triphylla grows well in a warm and sunny or shaded niche, but cannot grow in full shade.
Plants are hardy to about −20 °C. Slugs have been known to destroy its young growth or mature plants. Adenophora triphylla grows wild in mountains and meadows, but is cultivated; the seed can be sown in spring and germinates in 1–3 months. At that time, it needs a temperature of about 10 °C, it can be planted out into a permanent positions. Adenophora triphylla roots contain chemical compounds that are triterpenes. In Korea, A. triphylla is traditionally used for sputum and bronchial catarrh. It is believed to have antifungal and cardiotonic effects
Syed Nasir Ali Rizvi, s/o Brig. Gen. was a Pakistani politician. He started his education at Govt. High School of Kehror Pacca, he continued at Multan Convent, Mayfair School of Calcutta and completed his bachelors from Forman Christian College, Lahore. Following his father, after completing his studies, he joined Kakool, he resigned from the Army in 1957 and began his political career in 1958. He joined Maader-e-Millat Mohtarma Fatima Jinnah in her campaign against the dictator Ayub Khan, he was elected General Secretary of the Multan Council, Muslim League in 1965. He was the only person to contest elections against the group of Nawab of Kalabagh, he was elected as a Member of Multan Divisional Council. He was the only member on the opposition benches in the Multan Divisional Council. In 1968, he joined, he played a major role formation of Pakistan People's Party with Bhutto. In 1970, he won by 63,000 votes. In 1971, he went to Dhaka with Bhutto for talks with Sheikh Mujib-ur-Rehman. In 1972, he again accompanied Bhutto for talks with the Soviets.
That year he became member of Rural Development Board and Telegraph Committee. He addressed the students of University of Pennsylvania and represented Pakistan in the United Nations. In 1973, he accompanied Bhutto to Iran on the invitation of the Shah of Iran, he became joined the Sindh TAS Committee. In 1974, he became Secretary General of Punjab, he became a member of Punjab Small Industries Corporation and the Federal Member of Organizing Committee of PPP. In 1976 he became the Secretary Deputy Secretary General of PPP, Pakistan, he became Federal Minister for Housing, Works & Urban Development, which led him to play a major role in the development of Pakistan. He played a major role in the construction of the Faisal Mosque, constructed during his ministry, he traveled with Bhutto for discussions to China. He represented Pakistan at the Canada Habitat Conference, presided over the conference, he led a delegation to Sweden for talks, went to the UK to address overseas Pakistanis on the subject of the race riots of the time.
In 1978 the Assemblies were dissolved and martial law was declared. Rizvi's political career paused for 15 years. In 1993, after the dictatorship had been cleared, he lost due to severe vote rigging, he died in a car accident on the 5 February 2000. He was a signatory to the Constitution of Pakistan, he was awarded the highest Saudi civilian award