In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1 or e depending on the context. If the group operation is denoted * then it is defined by e * e = e.
The trivial group should not be confused with the empty set (which has no elements, and lacking an identity element, cannot be a group).
Given any group G, the group consisting of only the identity element is a trivial group and being a subgroup of G is called the trivial subgroup of G.
The trivial group is cyclic of order 1; as such it may be denoted Z1 or C1.
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
Rowland, Todd and Weisstein, Eric W., "Trivial Group" from MathWorld.
Shawn Corey Carter (born December 4, 1969), better known by his stage name Jay-Z, is an American rapper, songwriter, record producer, entrepreneur, and occasional actor. He is one of the most financially successful hip hop artists and entrepreneurs in America, having a net worth of over $450 million as of 2011.He has sold approximately 50 million albums worldwide, while receiving fourteen Grammy Awards for his musical work, and numerous additional nominations. He is consistently ranked as one of the greatest rappers of all-time. He was ranked #1 by MTV in their list of The Greatest MCs of All-Time in 2006. Two of his albums, Reasonable Doubt (1996) and The Blueprint (2001) are considered landmarks in the genre with both of them being ranked in Rolling Stone magazine's list of the 500 greatest albums of all time. Blender included the former on their 500 CDs You Must Own Before You Die.
Jay-Z co-owns the 40/40 Club, is part-owner of the NBA's Brooklyn Nets and is also the creator of the line Rocawear. He is the former CEO of Def Jam Recordings, one of the three founders of Roc-A-Fella Records, and the founder of Roc Nation. As an artist, he holds the record for most number one albums by a solo artist on the Billboard 200 with eleven. Jay-Z also has had four number ones on the Billboard Hot 100, one as lead artist.
Dwayne Michael Carter, Jr. (born September 27, 1982), better known by his stage name Lil Wayne, is an American rapper. At the age of nine, Lil Wayne joined Cash Money Records as the youngest member of the label, and half of the duo, The B.G.'z, with B.G.. In 1997, Lil Wayne joined the group Hot Boys, which also included rappers Juvenile, B.G., and Young Turk. Hot Boys debuted with Get It How U Live! that year. Lil Wayne gained most of his success with the group's major selling album Guerrilla Warfare, released in 1999. Also in 1999, Lil Wayne released his Platinum debut album Tha Block Is Hot, selling over one million copies in the U.S.
Although his next two albums Lights Out (2000) and 500 Degreez (2002) were not as successful (only reaching Gold status), Lil Wayne reached higher popularity in 2004 with Tha Carter, which included the single "Go D.J." Wayne also appeared on the Destiny's Child top ten single "Soldier" that year. In 2005, the sequel to Tha Carter, Tha Carter II, was released. In 2006 and 2007, Lil Wayne released several mixtapes and appeared on several popular rap and R&B singles. His most successful album, Tha Carter III, was released in 2008 and sold over 1 million copies in the U.S. its first week of release. It included the number-one single "Lollipop" featuring Static Major. It also includes the singles "A Milli" and "Got Money" featuring T-Pain and won the Grammy Award for Best Rap Album.