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Normal subgroup

In abstract algebra, a normal subgroup is a subgroup, invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng−1 ∈ N for all g ∈ G and n ∈ N; the usual notation for this relation is N ◃ G. Normal subgroups are important because they can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. A subgroup N of a group G is called a normal subgroup of G; the usual notation for this relation is N ◃ G, the definition may be written in symbols as For any subgroup N of G, the following conditions are equivalent to N being a normal subgroup of G. Therefore, any one of them may be taken as the definition: Any two elements commute regarding the normal subgroup membership relation: ∀g, h ∈ G, gh ∈ N ⇔ hg ∈ N.

The image of conjugation of N by any element of G is a subset of N: ∀g ∈ G, gNg−1 ⊆ N.' The image of conjugation of N by any element of G is N: ∀g ∈ G, gNg−1 = N. ∀g ∈ G, gN = Ng. The sets of left and right cosets of N in G coincide; the product of an element of the left coset of N with respect to g and an element of the left coset of N with respect to h is an element of the left coset of N with respect to gh: ∀x, y, g, h ∈ G, if x ∈ gN and y ∈ hN xy ∈ N. N is a union of conjugacy classes of G: N = ⋃g∈N Cl. N is preserved by inner automorphisms. For all n ∈ N and g ∈ G, the commutator = n − 1 g − 1 n g is in N. There is some group homomorphism G → H whose kernel is N; the trivial subgroup consisting of just the identity element of G and G itself are always normal subgroups of G. If these are the only normal subgroups G is said to be simple; the translation group is a normal subgroup of the Euclidean group in any dimension. This means: applying a rigid transformation, followed by a translation and the inverse rigid transformation, has the same effect as a single translation.

By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating rotating about the origin, translating back will not fix the origin and will therefore not have the same effect as a single rotation about the origin. Every subgroup N of an abelian group G is normal. A group, not abelian but for which every subgroup is normal is called a Hamiltonian group; the center of a group is a normal subgroup. The commutator subgroup is a normal subgroup. More any characteristic subgroup is normal, since conjugation is always an automorphism. In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal. If H is a normal subgroup of G, K is a subgroup of G containing H H is a normal subgroup of K. A normal subgroup of a normal subgroup of a group need not be normal in the group; that is, normality is not a transitive relation.

The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group; the two groups G and H are normal subgroups of their direct product G × H. If the group G is a semidirect product G = N ⋊ H N is normal in G, though H need not be normal in G. Normality is preserved under surjective homomorphisms, i.e. if G → H is a surjective group homomorphism and N is normal in G the image f is normal in H. Normality is preserved by taking inverse images, i.e. if G → H is a group homomorphism and N is normal in H the inverse image f -1 is normal in G. Normality is preserved on taking direct products, i.e. if N 1 ◃ G 1 and N 2 ◃ G 2 N 1 × N 2 ◃ G 1 × G 2. Every subgroup of index 2 is normal. More a subgroup, H, of finite index, n, in G contains a subgroup, K, normal in G and of index dividing n! called the normal core. In particular, if p is the smallest prime dividing the order of G every subgroup of index p is normal.

The fact that normal subgroups of G are the kernels of group homomorphisms defined on G accounts for some of the importance of normal subgroups. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, a group is imperfect if and only if

Helianthus salicifolius

Helianthus salicifolius is a North American species of sunflower known by the common name willowleaf sunflower. It is native to the central United States in the Great Plains and Ozark Plateau. There are a few reports of scattered populations in the Northeast and Midwest parts of the country, but these appear to be escapes from cultivation. Helianthus salicifolius grows in limestone prairies, it is a perennial herb up to 250 cm tall. Leaves are long but narrow, up to 21 cm long, but more than 1.2 cm wide. One plant produces 6-15 flower heads, each containing 10-20 yellow ray florets surrounding 50 or more red disc florets. Photo of herbarium specimen at Missouri Botanical Garden, collected in Missouri in 1992

Louisa Leaman

Louisa Leaman is a London-based author. Her debut novel, The Perfect Dress, published in October 2019 by Transworld Books, is an uplifting contemporary romance about vintage wedding dresses. Rights have been sold to the US, Germany and Spain, she is working on her second novel, Meant To Be, which will be published in October 2020. As well as writing novels, Louisa writes for the Victoria & Albert Museum website. Early life Louisa grew up in Loughton, Essex, she was educated at Bancroft's School, Woodford Green attended Leeds University, where she studied Art History followed by a PGCE teaching qualification in Secondary Art & Design. Career In 2004 Louisa won a writing competition in the Times Educational Supplement; this led to a publishing deal with Continuum International Publishing), for whom she wrote five teaching/behaviour management guides. As an experienced teacher and behaviour consultant, she delivered teacher training throughout the UK based on her book Managing Very Challenging Behaviour.

Louisa has written for Hachette Children's Books, with titles including The Garden and Born Free Elephant Rescue: The True Story of Nina and Pinkie, based on The Born Free Foundation's dramatic rescue of orphaned elephants. The book describes the story of Nina the African elephant, released into the wild after decades of captivity. Nina's story was televised by the BBC, in a documentary featuring British actor, Martin Clunes, called Born to be Wild. Louisa has contributed to many journals and newspapers, including The Guardian, The Observer, The Independent, she wrote a weekly column for the Times Education Supplement Magazine and a column for the Guardian-Series newspaper about motherhood. Leaman, L.. Managing Very Challenging Behaviour. Continuum International Publishing. ISBN 0-8264-8539-1 Leaman, L.. Classroom Confidential. Continuum International Publishing. ISBN 0-8264-8541-3 Leaman, L.. The Naked Teacher. Continuum International Publishing. ISBN 0-8264-8540-5 Leaman, L.. Dictionary of Disruption.

Continuum International Publishing. ISBN 978-0-8264-9466-5 Leaman, L.. The Perfect Teacher. Continuum International Publishing. ISBN 978-0-8264-9787-1 Louisa_Leaman

2018 Memphis Tigers football team

The 2018 Memphis Tigers football team represented the University of Memphis in the 2018 NCAA Division I FBS football season. The Tigers played their home games at the Liberty Bowl Memorial Stadium in Memphis and competed in the West Division of the American Athletic Conference, they were led by third-year head coach Mike Norvell. They finished the season 8–6, 5–3 in AAC to finish in a three-way tie for the West Division championship. After tie-breakers, they represented the West Division in the AAC Championship Game where they lost to East Division champion UCF, they were invited to the Birmingham Bowl. The Tigers finished the 2017 season 7 -- 1 in AAC play to be champions of the West Division, they represented the West Division in The American Championship Game where they lost to East Division champions UCF. They were invited to the Liberty Bowl. Listed in the order that they were released The AAC media poll was released on July 24, 2018, with the Tigers predicted to win the AAC West Division.

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Formaggio di Fossa

Formaggio di Fossa is a cheese from Sogliano al Rubicone in the Emilia-Romagna region of Italy. The cheese's name, which means "cheese of the pit", is derived from the process of ripening the cheese in special pits dug in tuff rock; the cheese is produced in the areas between the Rubicon and Marecchia river valleys. In 2009 formaggio di fossa was granted Denominazione di Origine Protetta status, the Italian equivalent of protected designation of origin. Fossa cheese is made with cow's milk, or a mixture of the two; the cheese matures around 30 days before being placed in the "fossa", a pit dug into the ground and lined with straw. The pit is prepared by burning straw inside to sterilize the space; the cheese is wrapped in cloth bags and placed in the pit, closed off while the cheese matures for an additional 80 to 100 days. The sealing of the pit limits the oxygen available to the cheese, enabling a process of anaerobic fermentation. After being removed from the pit, the cheese is allowed to ripen for an additional three months.

The technique of making formaggio di fossa dates back to the 15th century. Ambra di Talamello – a type of Formaggio di Fossa List of cheeses

Forsythia × intermedia

Forsythia × intermedia, or border forsythia is an ornamental deciduous shrub of garden origin. The shrub grows to 3 to 4 metres high; the opposite leaves turn yellowish or purplish in the autumn before falling. The bright yellow flowers are produced on one- to two-year-old growth and may be solitary or in racemes from 2 to 6; the hybrid is thought to be a cross between Forsythia viridissima and F. suspensa var. fortunei. A plant of seedling origin was discovered growing in the Göttingen Botanical Garden in Germany by the director of the forestry botanical garden of the Royal Prussian Academy of Forestry in Münden, H. Zabel in 1878. Zabel formally described and named the hybrid in Gartenflora in 1885, it was introduced to the Arnold Arboretum in the United States in 1889. The hybrid is drought-tolerant. Like some other forsythias it is one of the earliest shrubs to flower. Well adapted to temperature changes, it blooms with bright yellow flowers, that are noticeable in twilight, it is one of several forsythia species that are cultivated in gardens and parks.

Cultivars include:-'Arnold Dwarf' - low-growing with pale yellow flowers'Beatrix Farrand' - a floriferous cultivar'Gold Tide' - floriferous, with deep yellow autumn colour'Karl Sax' - deep yellow flowers with orange lines in the throat. Introduced by the Arnold Arboretum in 1960.'Lynwood' - large flowers with broad petals ’Lynwood Variety’agm'Spectabilis"Spring Glory' - purple-tinged foliage in autumn'Variegata' - leaves with contrasting cream edges Week End=’Courtalyn’agm The first dirigent protein was discovered in Forsythia intermedia. This protein has been found to direct the stereoselective biosynthesis of -pinoresinol from coniferyl alcohol monomers