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Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. Group theory is central to many areas of pure and applied mathematics and the classification theorem has been called one of the great achievements of modern mathematics; the proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers; the Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

Gorenstein and Solomon are publishing a simplified and revised version of the proof. The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group. Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups; the completed proof of the classification was announced by Aschbacher after Aschbacher and Smith published a 1221-page proof for the missing quasithin case. Gorenstein wrote two volumes outlining the low rank and odd characteristic part of the proof, Michael Aschbacher, Richard Lyons, Stephen D. Smith et al. wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows: The simple groups of low 2-rank are groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.

The simple groups of small 2-rank include: Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit–Thompson theorem. Groups of 2-rank 1; the Sylow 2-subgroups are either cyclic, easy to handle using the transfer map, or generalized quaternion, which are handled with the Brauer–Suzuki theorem: in particular there are no simple groups of 2-rank 1. Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, wreathed, or a Sylow 2-subgroup of U3; the first case was done by the Gorenstein–Walter theorem which showed that the only simple groups are isomorphic to L2 for q odd or A7, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L3 or U3 for q odd or M11, the last case was done by Lyons who showed that U3 is the only simple possibility. Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem; the classification of groups of small 2-rank ranks at most 2, makes heavy use of ordinary and modular character theory, never directly used elsewhere in the classification.

All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 MacWilliams showed that its Sylow 2-subgroups are connected, the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. A group is said to be of component type if for some centralizer C of an involution, C/O has a component; these are more or less the groups of Lie type of odd characteristic of large rank, alternating groups, together with some sporadic groups. A major step in this case is to eliminate the obstruction of the core of an involution; this is accomplished by the B-theorem, which states that every component of C/O is the image of a component of C. The idea is that these groups have a centralizer of an involution with a component, a smaller quasisimple group, which can be assumed to be known by induction. So to classify these groups one takes every central extension of every known finite simple group, finds all simple groups with a centralizer of involution with this as a component.

This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, the groups of Lie type of and odd characteristic are quite different. A group is of characteristic 2 type if the generalized Fitting subgroup F* of every 2-local subgroup Y is a 2-group; as the name suggests these are the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, whe

Brachytrachelopan

Brachytrachelopan is a short-necked sauropod dinosaur from the Late Jurassic of Argentina. The holotype and only known specimen was collected from an erosional exposure of fluvial sandstone within the Cañadón Calcáreo Formation on a hill 25 kilometres north-northeast of Cerro Cóndor, Chubut Province, in west-central Argentina, South America. Though incomplete, the skeletal elements recovered were found in articulation and include eight cervical, twelve dorsal, three sacral vertebrae, as well as proximal portions of the posterior cervical ribs and all the dorsal ribs, the distal end of the left femur, the proximal end of the left tibia, the right ilium. Much of the specimen was lost to erosion many years before its discovery; the type species is Brachytrachelopan mesai. The specific name honours Daniel Mesa, a local shepherd who discovered the specimen while searching for lost sheep; the genus name translates as "short-necked Pan". This taxon's short neck is evidence that this lineage specialized to fill an ecological niche not exploited by other members of this infraorder.

Small for a sauropod, Brachytrachelopan measured less than 10 metres in length. Rauhut et al. note that the high degree of fusion present between the preserved neural arches and their respective centra, as well as fusion between the sacral centra, sacral neural arches, sacral neural spines is evidence that the holotype does not represent a juvenile animal. Hence, the small body size is not a relic of ontogeny. Rauhut et al. diagnose Brachytrachelopan as differing from all other sauropods in the following respects: "...individual cervical vertebrae being as long as, or shorter in anteroposterior length than, high posteriorly. Further apomorphies...include a pronounced, pillar-like centropostzygapophyseal lamina in the cervical vertebrae, a pronounced anterior inclination in the mid-cervical neural spines, with the tip of the spine extending beyond the anterior end of the centrum, anterior dorsal neural spines one to six with vertical bases and anteriorly flexed tips." Brachytrachelopan belongs to Sauropoda and Neosauropoda from the group of Diplodocoidea and family Dicraeosauridae.

Following a cladistic analysis of 27 sauropod taxa and 154 anatomical characters, Rauhut et al. assigned Brachytrachelopan to the Dicraeosauridae, proposing that, within this clade, it should be considered to have a sister group relationship to the Late Jurassic African taxon Dicraeosaurus, instead of to Amargasaurus from the Lower Cretaceous of South America. Rauhut et al. conclude this is indicative of a rapid evolutionary radiation and dispersal of the Dicraeosauridae following the separation of the continents of the Southern and Northern Hemispheres during the latest Middle Jurassic. The following cladogram by Tschopp and colleagues shows the presumed relationships between members of the Dicraeosauridae: Rauhut and colleagues in 2005 noted that the tendency towards shorter-necks seen in dicraeosaurids, most evident in Brachytrachelopan, runs counter to the lengthening of the neck seen in most sauropod lineages and indicates that this group of sauropods was "progressively adapting for low browsing and might have been specialized on specific food sources, as has been suggested for Amargasaurus and Dicraeosaurus."

Moreover, the morphology of the cervical neural arches in Brachytrachelopan would have restricted dorsal flexion of neck and most indicates that this sauropod was specialized to a diet of plants "growing at heights of between about 1 and 2 m." Rauhut and colleagues suggested that diet may have been a limiting factor in body size among dicraeosaurids, that this may have placed them in the same ecological niche as "large low-browsing iguanodontian ornithopods." Such large iguanodontians are absent from the Late Jurassic Gondwanan sediments that have produced all known fossils of dicraeosaurids, while they are abundant in similar ecosystems of the same age in North America, where dicraeosaurids are absent. This may indicate that large iguanodontians and dicraeosaurids were ecological analogs, resulting from parallel evolution in two distantly related dinosaurian lineages. National Geographic

Christopher Wight

Christopher Wight is a former cricketer from the Cayman Islands. A wicket-keeper, he played for the Cayman Islands national cricket team from 2000 to 2004. Wight made his debut for the Cayman Islands in August 2000 when he played against the USA and Bermuda in the Americas Championship at the Maple Leaf Cricket Club in King City, Ontario. In the year, he played in four List A matches as part of the Red Stripe Bowl in Antigua, his other two international tournaments were the 2002 Americas Championship in Buenos Aires and the 2004 Americas Championship in Bermuda. Christopher's parents are Marguerite Wight, he has five sisters. He came from a cricketing family, his twin brother David played cricket for the Cayman Islands, as did two other brothers. His grandfather Oscar played for British Guiana, his great-uncle Vibart played Test cricket for the West Indies

Idle Cure

Idle Cure was an arena rock band from Long Beach, California. The Encyclopedia of Contemporary Christian Music calls their sound "the best example of cloning a sound for Christian markets", likening it to that of Def Leppard's Pyromania, they targeted a youthful audience, distinguished by overtly evangelical religious lyrics. CCM magazine reported that all original members of the band had been in mainstream bands prior to the formation of Idle Cure, their first four albums have been reissued as compilations by KMG Records. 1986: Idle Cure 1988: Tough Love 1990: 2nd Avenue 1991: Inside Out 1992: Breakaways 1994: Eclipse 1998: Idle Cure/2nd Avenue 2000: Tough Love/Inside Out Mark Ambrose - guitar Pete Lomakin - keyboard, vocals Steve Shannon - vocals Clark Edmond - drums Glenn Pearce - guitar Chuck King - guitar "Toys In The Band". CCM Magazine. 11: 16. July 1988. ISSN 1524-7848. Profile at Jesus Freak Hideout Profile at the Christian Music Archive

Ferencvárosi TC (women's handball)

Ferencvárosi Torna Club is a Hungarian women's handball team from Budapest, part of the multi-sports club Ferencvárosi TC. Nicknamed Fradi, the team plays in the top level championship in Hungary, they are one of the most successful clubs in the country, having won eleven Hungarian championship and as many Hungarian cup titles. FTC enjoy a good reputation in continental competitions: they lifted the EHF Cup Winners' Cup trophy in 1978, 2011, 2012, they were crowned as the EHF Cup winners in 2006; the team reached the finals of the EHF Champions League two times, they fell short in both occasions. The current name of the club is FTC-Rail Cargo Hungaria due to sponsorship reasons; the following table shows in detail Ferencvárosi TC kit manufacturers and shirt sponsors by year: Squad for the 2019–20 season Transfers for the 2020-21 season Head Coach: Gábor Elek Assistant Coach: Attila Kovács Goalkeeping Coach: Norbert Duleba Fitness Coach: Tord Ellingsen Club Doctor: Attila Pavlik, MD Physiotherapist: Dorottya Vajay-Gazsó Nemzeti Bajnokság I Champions: 1966, 1968, 1969, 1971, 1993–94, 1994–95, 1995–96, 1996–97, 1999–00, 2001–02, 2006–07, 2014–15 Runners-up: 1963, 1967, 1970, 1972, 1973, 1976, 1977, 1978, 1992–93, 1998–99, 2000–01, 2002–03, 2005–06, 2008–09, 2011–12, 2012–13, 2013–14, 2015–16, 2016–17, 2017–18 Third place: 1974, 1975, 1979, 1980, 1987, 1997–98, 2003–04, 2004–05, 2007–08, 2010–11Magyar Kupa Winners: 1967, 1970, 1972, 1977, 1992–93, 1993–94, 1994–95, 1995–96, 1996–97, 2000–01, 2002–03, 2016–17 Finalist: 1963, 1973, 1978, 1986, 1997–98, 1998–99, 2006–07, 2009–10, 2012–13, 2013–14, 2014–15 EHF Champions League: Runners-up: 1970-71, 2001–02 Semifinalists: 1996, 1997, 2001EHF Cup Winners' Cup: Winners – record: 1977-78, 2010-11, 2011-12 Runners-up: 1978-79, 1993-94 Semifinalists: 2007, 2015EHF Cup: Winners: 2005-06 Semifinalists: 2004-05EHF Champions Trophy: Third Placed: 2002 Fourth Placed: 2006 Baia Mare Champions Trophy: Second Placed: 2014 As of 7 September 2019.

Seasons in Nemzeti Bajnokság I: 62 Participations in Champions League: 24x Participations in EHF Cup: 4x Participations in Cup Winners' Cup: 9x Endre Balogh Gyula Elek Longest serving coach in Ferencvárosi TC's history Gyula Elek and András Németh Károly Konkoly Mária Berzsényi Pál Hoffmann András Németh Most honours won with Ferencvárosi TC Gyula Zsiga Gábor Elek Son of former coach Gyula Elek. Ferencvárosi TC II is the junior team of Ferencvárosi TC women's handball club, they compete in the second-tier league in Hungary. Although they play in the same league system as their senior team, rather than a separate league, they are ineligible for promotion to the Nemzeti Bajnokság I, since junior teams cannot play in the same division as their senior side. Ferencvárosi TC Official Website Club page on the European Handball Federation website Ferencvárosi TC on Facebook

Primary (band)

Primary were an Australian techno rock band which formed in 1995 the Fonti brothers: Jamie on keyboards and Sean on bass guitar, Connie Mitchell on lead vocals. According to Australian musicologist, Ian McFarlane, the group were "Dominated by South African-born's hyperactive and full-frontal vocals, with thunderous electronic rock underpinning the music, Primary sounded like a techno Skunk Anansie. Jamie Fonti coined the phrase'Hybrid Electronica Rock' in order to describe the band's sound." The group released two albums, This Is the Sound and Watching the World. They disbanded late in 2003. Primary formed in 1995 in Sydney as a techno rock group by the Fonti brothers Jamie on keyboards and Sean on bass guitar, Connie Mitchell on lead vocals; the Fontis had played in My Heart Bleeds for a mid-1980s punk, hardcore band. Sean was in Massappeal, Caligula from and Def FX; the Fonti brothers had met Mitchell in a recording studio in 1994, while they recorded demo tracks for Caligula. When that project finished Jamie and Mitchell started writing tracks together while Sean was a member of Def FX.

Primary's first recording, a five-track extended play, Vicious Precious, produced by Paul McKercher and Ollie J, was released in March 1998, featured Paul Wheeler on drums. Bousfield left the band in 1999. Nick Launay produced the band's first full-length album, This Is the Sound, which appeared in June that year on WEA/Warner, it peaked in the ARIA Albums Chart top 40. Australian musicologist, Ian McFarlane, felt it was "an accomplished album that boasted state-of-the-art production values, a batch of reverberating tracks." He described how the group were "Dominated by South African-born's hyperactive and full-frontal vocals, with thunderous electronic rock underpinning the music, Primary sounded like a techno Skunk Anansie. Jamie Fonti coined the phrase'Hybrid Electronica Rock' in order to describe the band's sound."Jason Howard joined Primary on lead guitar and made his performance debut in April 1999. They became known for their energetic live shows, with Mitchell's on-stage presence and costumes a talking point.

Their second album, Watching the World, was released on 28 May 2001. Australian music journalist, Ed Nimmervoll, declared it to be his Album of the Week: "they kept the focus on the songs. In the end Primary offer music with many layers of interest.'s lyrics are a strong counterpoint to the depth of the band's music."The band announced on their website on 24 April 2002 that they were recording demos for a proposed third album, but it was not released. They played a gig in November 2003 at the Annandale Hotel. Mitchell joined Sneaky Sound System in 2004 ahead of their self-titled album; this Is the Sound – AUS #33 Watching the World "Vicious Precious/Brazilian" – AUS #84 "Young /This Is the Sound" – AUS #89 "Supposed to Be Here" – AUS #72 "In Your Hands" "Not for Me" "Get Amongst It" – Used by TV Channel Ten in advertisements such as Sports Tonight, The Shield and Cops