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Conjugacy class

In mathematics group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g–1ag. This is an equivalence relation. Members of the same conjugacy class cannot be distinguished by using only the group structure, therefore share many properties; the study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element. Functions that are constant for members of the same conjugacy class are called class functions. Let G be a group. Two elements a and b of G are conjugate. One says that b is a conjugate of a and that a is a conjugate of b. In the case of the group GL of invertible matrices, the conjugacy relation is called matrix similarity, it can be shown that conjugacy is an equivalence relation and therefore partitions G into equivalence classes. The equivalence class that contains the element a in G is Cl = and is called the conjugacy class of a.

The class number of G is the number of distinct conjugacy classes. All elements belonging to the same conjugacy class have the same order. Conjugacy classes may be referred to by describing them, or more by abbreviations such as "6A", meaning "a certain conjugacy class of order 6 elements", "6B" would be a different conjugacy class of order 6 elements. In some cases, conjugacy classes can be described in a uniform way; the symmetric group S3, consisting of the 6 permutations of three elements, has three conjugacy classes: no change transposing two a cyclic permutation of all three These three classes correspond to the classification of the isometries of an equilateral triangle. The symmetric group S4, consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders: 4 no change interchanging two a cyclic permutation of three a cyclic permutation of all four interchanging two, the other two The proper rotations of the cube, which can be characterized by permutations of the body diagonals, are described by conjugation in S4.

In general, the number of conjugacy classes in the symmetric group Sn is equal to the number of integer partitions of n. This is because each conjugacy class corresponds to one partition of into cycles, up to permutation of the elements of. In general, the Euclidean group can be studied by conjugation of isometries in Euclidean space; the identity element is always the only element in its class, Cl = If G is abelian gag−1 = a for all a and g in G. If two elements a and b of G belong to the same conjugacy class they have the same order. More every statement about a can be translated into a statement about b = gag−1, because the map φ = gxg−1 is an automorphism of G. See the next property for an example. If a and b are conjugate so are their powers ak and bk, thus taking kth powers gives a map on conjugacy classes, one may consider which conjugacy classes are in its preimage. For example, in the symmetric group, the square of an element of type is an element of type, therefore one of the power-up classes of is the class.

An element a of G lies in the center Z of G if and only if its conjugacy class has only one element, a itself. More if CG denotes the centralizer of a in G, i.e. the subgroup consisting of all elements g such that ga = ag the index is equal to the number of elements in the conjugacy class of a. Take σ ∈ S n and let m 1, m 2... M s be the distinct integers which appear in the cycle type of σ. Let k i be the number of cycles of length m i in σ for each i = 1, 2... s. The number of conjugates of σ is: n

Triple F Racing

Triple F Racing was an Australian motor racing team that competed in the International V8 Supercars Championship between 2009 and 2013. The team was based on the Gold Coast in Queensland but continued to have strong links with the team's original home in Perth. In its final two years its Racing Entitlement Contract was leased to Dick Johnson Racing. At the end of 2012 the REC was returned to the issueing organisation, V8 Supercars Australia, formally ending its participation in V8 Supercar; the team was formed by the West Australian Fiore racing family, with the team name referencing the three Fiore brothers Dean and Todd. The team raced out of Western Australia for many years running in state level series in Formula Ford and Saloon Car racing. After the collapse of Team Kiwi Racing as a V8 Supercar team in early 2009, Triple F Racing was created to keep Dean Fiore, racing the Holden VE Commodore utilising the Team Kiwi Racing REC, but the car was in essence a third car for the Paul Morris Motorsport team.

Triple F Racing formally took over the car as of the Winton round of the 2009 series. In keeping with results achieved by second and/or third Paul Morris Motorsport supported vehicles, results were modest. For the 2010 season, a Ford FG Falcon was acquired from Paul Cruikshank Racing and the team moved into their own workshop on the Gold Coast. Dean Fiore moved up the grid in terms of performance with the model car; the former VE Commodore was used by Paul and Todd Fiore in the second-tier Development Series until round 5 where the team debuted an ex Triple 8 BF Falcon, used for the rest of the season. Michael Patrizi was the teams co-driver in the endurance races. In 2011 season; the team continued with the FG Falcon, while the BF Falcon was used by the team in the V8 Supercar Development Series. Patrizi was retained as the teams endurance driver. In 2012, the REC, Fiore and car moved to become a third Dick Johnson Racing entry. Matt Halliday co-drove in the endurance events. In 2013 the REC remained with Dick Johnson with Jonny Reid and Chaz Mostert driving a new Ford FG Falcon, with Dean Fiore driving a Holden VF Commodore for Lucas Dumbrell Motorsport.

At the end of the season the REC was returned by Dick Johnson Racing, after unsuccessfully trying to raise sponsorship for a 2014 campaign, was returned to V8 Supercars Australia. In addition to the V8 Supercar team, the original West Australian based Triple F Racing continued to support Paul Fiore racing Holden Commodore in Saloon Cars, Todd Fiore in Formula Ford during the 2009 season. Official website

Right of abode (United Kingdom)

The right of abode is an immigration status in the United Kingdom that gives a person the unrestricted right to enter and live in the UK. It was introduced by the Immigration Act 1971 which went into effect on 1 January 1973; this status is held by British citizens, certain British subjects, as well as certain Commonwealth citizens with specific connections to the UK before 1983. Since 1983, it is not possible for a person to acquire this status without being a British citizen; the right of abode is the most common immigration status in the UK due to its association with British citizenship. However, it should not be confused with the indefinite leave to remain, another form of long-term residency status in the UK, more comparable to other countries' permanent residence status. All individuals who have the right of abode in the UK enjoy the following rights and privileges: an unconditional right to enter, live and study in the United Kingdom an entitlement to use the UK/EEA/Switzerland immigration channel at United Kingdom ports of entry an entitlement to apply for United Kingdom social security and welfare benefits a right to vote and to stand for public office in the United Kingdom.

Unlike the indefinite leave to remain, a person's right of abode is valid for life and will not lapse regardless how long they stay outside the UK and cannot be revoked unless they lose their Commonwealth or British citizenship, or, for Commonwealth citizens, when the Home Secretary deemed it necessary for "the public good". In comparison, a person's ILR status may be revoked if they had remained outside the UK for more than two years and they were refused a resident return visa. In addition, those with the right of abode who are not yet British citizens may apply for British citizenship by naturalisation. Children born in the United Kingdom, British Crown Dependencies and British Overseas Territories to those with the right of abode in the UK will be British citizens by birth automatically. Prior to the enactment of British Nationality Act 1981, right of abode in the UK was determined by a mix of one's connection with the UK and their nationality status; the Immigration Act 1971 limited the right to enter and live in the United Kingdom to certain subsets of Citizens of the United Kingdom and Colonies with ties to the UK itself.

The following two categories of persons had right of abode: Citizens of the United Kingdom and Colonies CUKCs who met any of the following requirements acquired right of abode between 1973 and 1983. Born in a part of the British Islands. Commonwealth citizensAny Commonwealth citizens who met one of the requirements below would acquire right of abode between 1973 and 1983. Born to or adopted by a CUKC parent born in the British Islands; the person they were married to must not have a living ex-partner who had acquired right of abode in the UK at any given time. Right of abode was limited to CUKCs and Commonwealth citizens, therefore certain people with connections to the UK were not eligible if they had a UK-born parent. For example, a person born to a UK-born CUKC mother and a non-Commonwealth citizen father in the United States would not acquire right of abode as they possessed neither CUKC status nor Commonwealth citizenship. However, the same person would obtain right of abode if born in Canada, due to Canada's membership in the Commonwealth when they would not be a CUKC or, since 1983, a British citizen.

That wording of the measure introduced minor confusion into the concept of the right of abode, when it limited right of abode through descent to a CUKC who had a parent who had CUKC status by "birth, naturalisation or.... Registration in the United Kingdom or in any of the islands" or a grandparent CUKC who "at the time of that birth or adoption so had it". Whether "so had it" referred to a grandparent, had CUKC status or CUKC status from the UK and islands was decided by courts to refer to the latter; as the right of abode on 31 December 1982 was necessary to become a British citizen on 1 January 1983 under the British Nationality Act 1981, this meant that failing to meet the interpretation of the right of abode test above resulted in no British citizenship through that route. The British Nationality Act 1981 modified the right of abode section of the Immigration Act 1971 to remove the wording at issue, although the former version still had effect for determinations of Brit

Mike McCormick (outfielder)

Myron Winthrop "Mike" McCormick was an American professional baseball player. He was an outfielder for the Cincinnati Reds, Boston Braves, Brooklyn Dodgers, New York Giants, Chicago White Sox and Washington Senators of Major League Baseball. McCormick was born in Angels Camp, stood 6 feet tall, weighed 190 pounds, threw and batted right-handed, he helped the Reds win the 1940 World Series, led the National League in sacrifice hits that season. Forty games into the 1942 season, McCormick was sidelined with a broken leg, he returned to the Reds for the 1943 season, but he was inducted into the military after only a few games. He missed the 1944 and 1945 seasons due to military service, he played on a military baseball team with Joe DiMaggio in Hawaii. After the Braves won the 1948 National League pennant, he was traded from the Braves to the Brooklyn Dodgers in December of that year; the Dodgers won the 1949 NL pennant. The next year, he was signed by the New York Giants, played for Oakland of the Pacific Coast League, had his contract purchased by the Chicago White Sox that June.

In 10 seasons he played in 748 games and had 2,325 at bats, 302 runs, 640 hits, 100 doubles, 29 triples, 14 home runs, 215 RBI, 16 stolen bases, 188 walks.275 batting average.330 on-base percentage.361 slugging percentage, 840 total bases and 72 sacrifice hits. Defensively, he recorded a.980 fielding percentage at all three outfield positions. In April 1976, McCormick was attending a game at Dodger Stadium, he died at a Los Angeles hospital. Career statistics and player information from Baseball-Reference, or Baseball-Reference Mike McCormick at Find a Grave

Kiara Ortega

Kiara Liz Ortega Delgado is a Puerto Rican dancer and beauty pageant titleholder, crowned Miss Universe Puerto Rico 2018. She represented Puerto Rico at the Miss Universe 2018 pageant held in Thailand, she finished in the Top 5 at Miss Universe 2018. Ortega was born in Puerto, she works for an organization named Centro Inés which provides different educational and support services for children and seniors to help develop a sense of independence and mastery of skills. On 20 September 2018, Ortega began her pageantry career representing Rincón in the Miss Universe Puerto Rico 2018 competition at Luis A. Ferré Performing Arts Center in San Juan, where she was crowned as Miss Universe Puerto Rico 2018, she succeeded outgoing Miss Universe Puerto Rico 2017 Danna Hernández. Ortega is the third woman with noticeably mixed features to win Miss Universe Puerto Rico after Alba Reyes in 2004 and Zuleyka Rivera in 2006. Ortega represented Puerto Rico at Miss Universe 2018 pageant in Bangkok, finishing in the Top 5 ending a 4-year drought for Puerto Rico and giving the Island its first Top 5 placement since 2009 when Mayra Matos finished as 4th Runner-Up.

In 2018 Kiara wore a costume to represent Puerto Rico which featured the Puerto Rican flag colors, three pairs of hands and a star for a headdress. Miss Universe Puerto Rico Official Website

Anthony Hamlett

Anthony Hamlett is an American mixed martial artist and fight promoter from Washington State. He is the head manager of Northwest Elite Fight Team, he has been a professional referee in the Northwest and referred the UFC. He has refereed over 3000 fights between pro, he is looked at as the top ref in MMA for the NW. Anthony Hamlett was in the US Air Force from March 1992- October 2000. Anthony Hamlett in the USWF/USA WRESTLING ALL-AMERICANS A three time All American for the USAF team ranked as high as 5th in nation after world team trials ADCC Championships 2000 March 1, 2 & 3rd 2000 in Abu Dhabi, UAE the -65 kg class. Anthony Hamlett Defeated Hiroyuki Abe, Joao Roque, Joe Gilbert. Anthony suffered a loss to Royler Gracie to place 4th at the Abu Dhabi World Championship after just one yr of pure grappling training coming over from wrestling HOOKnSHOOT Featherweight World Champion Professional MMA record for Anthony Hamlett from Sherdog