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Cyclic group

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.

For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group G of order n we have G =, where e is the identity element and gi = gj whenever i ≡ j. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.

Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.

To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z, with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.

If p is a prime Z/pZ is a finite field, is denoted Fp or GF. Fo

Hengyang County

Hengyang County is a county and the 5th most populous county-level division in the Province of Hunan, China. Located in the north of Hengyang City and the south east of Hunan province, the county is bordered to the north by Shuangfeng County, to the west by Shaodong County, to the south by the counties of Qidong and Hengdong and the districts of Zhengxiang and Zhuhui, to the east by Nanyue District and Hengshan County. Hengyang County covers 2,558.61 km2 with a population of 1,235,100. The county has 17 towns under its jurisdiction, the county seat is Xidu Town; the county was the first time named after Hengyang in history that Hengyang County was formed through the amalgamation of the three counties of Linzheng and Chong'an in 589 AD. After that the county was once again divided into three counties of Linzheng and Chong'an, the two counties of Xincheng and Chong'an were merged to the county of Linzheng, the county of Linzheng was turned back to the name of Hengyang in 732 AD; the provincial city of Hengyang was established from three townships and urban area of Hengyang County in January 1931.

Hengnan County was formed from the 1st, 2nd, 4th and 9th districts in the south eastern portion of the county in April 1952, the county seat was transferred to Xidu Town in July 1952. The county of Hengyang is located in the northwest of Hengyang City and the middle reaches of the Xiang River, it borders Nanyue District and Hengshan County to the east, is adjacent to Zhengxiang District, Shigu District and Hengnan County in the south, Qidong County and Shaodong County in the west, Shuangfeng County in the north. It is 74 kilometers from east to 55 kilometers from north to south; the mountainous part of the county is the extension of the Hengshan range, rolling down. Its terrain is high in the northwest and low in the southeast, with plains staggered. There are many plains on both sides of the rivers; the Zheng River merges to the river of Xiang at Shiguzui. It is convenient for water transportation with the Xiang Zheng River as the main channel. Mineral deposits in the county are placer gold, ceramic mud, kaolin, barite and zinc.

Hengyang County is located in the transitional zone between the rise of Wuling Mountains and the subsidence of Dongting Lake, the northern edge of Hengyang Basin. The layers of tertiary red rock is deposited in the center of the basin and is about 3,000 meters thick. A series of tenia fornicis on the east and west sides are centered around the central-south red basin with different types of structures. Hengyang has a subtropical monsoon climate with cool summers and warm winters, it is warm and humid; the annual precipitation is 1,452 mm, the annual average temperature is about 17.9 °C, the average temperature in January is 4.6 °C and in July is 30.3 °C. According to preliminary accounting of the statistical authority, the gross domestic product of Hengyang County in 2017 was 35,493 million yuan, up by 8.5 percent over the previous year. Of this total, the value added of the primary industry was 7,108 million yuan, up by 3.5 percent, that of the secondary industry was 12,656 million yuan, up by 7.4 percent and that of the tertiary industry was 15,729 million yuan, up by 12.2 percent.

The value added of the primary industry accounted for 20.03 percent of the GDP. The per capita GDP in 2017 was 33,115 yuan. Gouloufeng National Forest Park: a forest park at national level and one of AAA-rated tourist attractions in China with an area of 20.67 km2 located in Goulou Township. Wanyuan Lake Scenic Area: one of national water scenic areas located between the towns of Jiepai and Shishi. Longwangxia Ecological Tourist Resort: a recreational resort with whitewater rafting, rock climbing, honorable person CS and outward development located in Longwang Village of Jinlan Town. Xiangxi Cottage: Xiangxi Cottage is the former residence of Wang Chuanshan, one of the provincial heritage conservation units of Hunan, a building of southern Hunan in the late Ming and early Qing dynastys located in Xiangxi Village of Qulan Town. Oriental Manor: a scenic spot of rural tourism with an area of 45.33 hectares, one of the national aquatic health farms by the Ministry of Agriculture, one of provincial agricultural tourism demonstration sites of Hunan and a Five - Star leisure manor in Hunan located in Yingpo Village of Xidu Town.

Xia Minghan's Former residence: the Former Residence of Xia Minghan, an early leader of the Chinese revolution is a typical residence of southern Hunan in the Qing dynasty. Yishan Temple: the Yishan Temple, a famous temple in China with a history of over 1,700 years located in the north of Yishan Village in Shanqiao Town. Weaver Lake Water Scenic Area: or Zhinu Lake Water Scenic Area, one of national water scenic areas in China located Quanjing Village of Xinashan Town. Forest Ecological Park of Jiufen Mountain: a forestecological park located in Jiufeng Village of Xijiang Township. Danxia Landform Scenic Area of Shishi: a danxia landform scenic area located in Shishi Town, the south of Hengshan Mountains and north of Hengyang County

Six Flags Zhejiang

Six Flags Zhejiang is an upcoming theme park in Jiaxing, China, a partnership between Six Flags. It will feature original rides and attractions, along with attractions themed around Looney Tunes and Friends, Tuzki, among others. In addition to a theme park, the park will be accompanied by a Six Flags Hurricane Harbor water park. Six Flags Zhejiang will be located on the coast of Hangzhou Bay; the park is 45 miles from downtown Shanghai. As of 2020, construction on the park has came to a halt, as the Riverside Investment Group has been unable to pay their debts; the main area of the park, which will branch off to the park's other areas. The park's areas are based on other Six Flags theme parks as well as various aspects of the United States. Themed after Six Flags Magic Mountain in Valencia, California. Themed after Six Flags Great Adventure in Jackson, New Jersey. Themed after Six Flags Fiesta Texas in San Antonio, Texas. A beach-themed children's area, based on the Garfield franchise by Jim Davis.

A seaside area connected to Garfield Beach. A water park, similar to other Six Flags Hurricane Harbor water parks in the US. Six Flags Official website

2009 VFL season

The 2009 Victorian Football League season was the 128th season of the Australian Rules Football competition. The premiership was won by the North Ballarat Football Club, which defeated the Northern Bullants by 23 points in the Grand Final on 25 September, it was North Ballarat's second consecutive premiership, the second in the club's history. At the end of the 2008 season, the Tasmanian Devils Football Club withdrew from the VFL and disbanded. AFL Tasmania, which operated the club, was focussed on re-establishing the Tasmanian Football League as a statewide competition in 2009, after an eight-year hiatus since the original statewide league's collapse at the end of 2000, having the Tasmanian VFL club competing for attention and players did not fit with this vision; as a result, the VFL was reduced to thirteen clubs. Additionally, two pairs of VFL-AFL reserves affiliations were altered: Melbourne ended its nine-year affiliation with Sandringham and entered a new affiliation with the Casey Scorpions.

St Kilda ended its eight-year affiliation with the Casey Scorpions and entered a new affiliation with Sandringham. As a result, the size of the VFL was reduced to thirteen teams: nine VFL-AFL affiliates, two AFL reserves team, two stand-alone VFL teams; the Jim'Frosty' Miller Medal was won by Nick Sautner for the ninth time in his career. Sautner kicked 71 goals for the season; the J. J. Liston Trophy was by Myles Sewell. Sewell finished ahead of Michael Barlow, second with 16 votes, James Podsiadly and Sam Iles, who were equal third with 15 votes; the Fothergill-Round Medal was won by Michael Barlow. Box Hill won the reserves premiership. Box Hill 16.18 defeated Sandringham 13.6 in the Grand Final, held as a curtain-raiser to the seniors second preliminary final on 20 September. The Bendigo Bombers could not play at their home ground Queen Elizabeth Oval after May because the surface was deemed unfit for VFL football. Bendigo's remaining home games were transferred to Windy Hill, the training ground of its AFL-affiliate Essendon.

The VFL introduced the "23rd man rule". Under the rule, the size of the playing squad was increased to twenty-three, provided the 23rd player was a top-age player in the TAC Cup, or an undrafted player who had played in the TAC Cup during the previous year; the rule was designed to provide additional senior football opportunities to promising juniors, improve the alignments between VFL clubs and their TAC Cup affiliates. List of VFA/VFL premiers Australian Rules Football Victorian Football League Australian Football League 2009 AFL season AFL Victoria website Official VFL website Unofficial VFL website

Radioaxiom: A Dub Transmission

Radioaxiom: A Dub Transmission is a collaborative album by Bill Laswell and Jah Wobble, released on September 18, 2001 by Axiom and Palm Pictures. Adapted from the Radioaxiom: A Dub Transmission liner notes. MusiciansAïyb Diengpercussion Hamid Drakedrums, tabla Sly Dunbar – drums Graham Haynescornet Karsh Kale – drums, tabla Bill Laswell – bass guitar, mixing Nils Petter Molværtrumpet Amina Claudine Myers – electric piano, organ Ejigayehu Shibabawvocals Tigist Shibabaw – vocals Nicky Skopelitis – guitar Jah Wobble – bass guitar, producerTechnical personnelJames Dellatacoma – assistant engineering Michael Fossenkempermastering James Koehnline – cover art Robert Musso – engineering Critical reaction to Radioaxiom was positive. Thom Jurek of Allmusic wrote that the album's subtitle, A Dub Transmition was somewhat misleading as the album was not influenced by dub reggae but taken on its own terms as a "mishmash of world musics and groove jazz it works well." He awarded Radioaxiom 4 stars out of a possible 5.

Radioaxiom – A Dub Transmission at Discogs

Gwen Davis

Gwen Davis is an American novelist, screenwriter, songwriter and poet. Davis has written eighteen novels, including the sexy bestseller The Pretenders, she has written about travel for the Wall Street Journal Europe, for online publications such as the Huffington Post, maintains a popular personal blog, Report from the Front, a blog reviewing Broadway theater productions, Will Blog for Broadway. Davis was born in Pittsburgh and grew up in New York City, in Manhattan, her parents were divorced. Her father, real estate developer Lew Davis served as mayor of Tucson, winning office in 1961, her parents' separation when she was five started a lifetime of gypsying. She attended Bryn Mawr College. In 1954, at the age of eighteen, she went to Paris to study music and sang in a nightclub there until she gave into her mother's pleas to return to the U. S, she moved to continued singing, performing at the Purple Onion. She obtained a master's degree in Creative Writing from Stanford University, she was part of the Hollywood social scene from the late 1950s, coming into contact with a wide range of celebrities and befriending Dennis Hopper and many others.

Some of her experiences inspired Naked in Babylon. She married businessman and producer Don Mitchell, with whom she had two children, a daughter and a son. One of the Mitchells' mocking Academy Awards parties was the subject of a Time magazine article in 1970, which mentioned some of the celebrities—Shirley MacLaine, Zsa Zsa Gabor, Lee Marvin and others—Davis and Mitchell counted among their friends, she scripted a famous movie, What a Way to Go, wrote a play produced on Broadway, The Best Laid Plans. Davis continues to write, she travels and has lived in Spain, Rome, Venice, New York and Hollywood. She has returned from living in Bali, Indonesia and is living between New York, Beverly Hills. Touching was not a bestselling novel. Davis spent twenty hours at Sandstone, a Topanga Canyon therapy center run by E. Paul Bindrim, known as the "father of Nude Psychotherapy". Bindrim, once nearly kicked out of the American Psychological Association, was known for holding what he called "nude marathons"—several clients were "placed in a warm pool for long sessions of touching and massaging and sometimes shouting or acting out rage".

After the novel was published, Bindrim sued Davis and Doubleday & Company for libel, on the grounds that it had defamed him. Davis claimed she had used her real-life experiences to inspire fiction, but that Bindrim was not the psychologist in her fictional story, did not resemble him—the character she depicted was overweight, looked like Santa Claus, had a Ph. D; when the case came to trial, bald and clean shaven, who held only a master's degree, had by gained weight, grown a white beard, been granted a Ph. D. from International College in Westwood, California. These changes made, he won his lawsuit against her publisher, Doubleday. Doubleday sued Davis for not disclosing her contractual agreement with Bindrim not to write about the psychotherapy event; because of the Bindrin precedent, American novelists in America became concerned about the possibility of lawsuits against writers who used real people as the basis for their fiction. Davis settled the lawsuit for an undisclosed amount. Naked in Babylon, 1960 Someone's in the Kitchen with Dinah, 1962 The War Babies, 1966 Sweet William, 1967 The Pretenders, 1969 Touching, 1971 Kingdom Come, 1972 Changes, 1973 The Motherland, 1974 How to Survive in Suburbia When Your Hearts in the Himalayas, 1976 The Aristocrats, 1977 Ladies in Waiting, 1979 Marriage, 1981 Romance, 1983 Silk Lady, 1986 The Princess and the Pauper: An Erotic Fairy Tale, 1989 Jade, 1991 Happy at the Bel Air, 1996 West of Paradise, 1998 Lovesong, 2000 Scandal, 2011 The Daughter of God, 2012 "Desperate Intruder," 1983 "Better Late Than Never," 1982 "What a Way to Go!," 1964 Davis appeared in Rich and Famous, 1981, as a party guest.

She was interviewed many times on The Tonight Show Starring Johnny Carson in 1971-1972, on David Frost and the Virginia Graham Show. Davis' Official Website, The Only Gwen Davis' Official Blog, Report From the Front Davis' Blog, Will Blog for Broadway