Algebraic Group

A group in modern algebra is a set G combined with a BinaryOperator, "*", often called the "group multiplication" where the following four axioms are satisfied -

1. Closure - if g, h are in G then g*h is in G.
2. Associative - if g*(h*j) = (g*h)*j for any g,h,j in G.
3. Identity - There is an element, e in G such that g*e = e*g = g for all g in G.
4. Inverse - For any g in G, there exists g^-1 in G such that g*(g^-1) = (g^-1)*g = e.

Examples include:

1. The set of integers with the operation of addition.
2. The set of positive rational numbers with the operation of multiplication.
3. The set with the single element 1 and the operation * defined so that 1*1=1 (The trivial group)
4. The set of all rigid movements of a square:

1. Flip the square vertically
2. Flop the square horizontally
3. Flip the square across the northeast-southwest diagonal
4. Flip the square across the northwest-southeast diagonal
5. Rotate the square 90 degrees clockwise
6. Rotate the square 180 degrees
7. Rotate the square 90 degrees counterclockwise
8. Leave the square alone

and the operation 'and then'. For example, combining items 5 and 2, we get 'rotate the square ninety degrees clockwise, and then flip it over horizontally'; the result of this operation is the same as item 3.)

Some non-examples include:

1. The set of positive integers with the operation of subtraction. (The operation is not closed, because 7-12 is not a positive integer.)
2. The set of all integers with the operation of subtraction. (The operation is not associative, since 7-(5-1) is not the same as (7-5)-1.)
3. The set of positive integers with the operation of addition. (There is no identity element)
4. The set of integers with the operation of multiplication. (The identity is 1, but the element 0 has no inverse.)

Note that the property g*h = h*g is not required. If this property holds for all g and h, the operation is said to be commutative and the group is an AbelianGroup. The rigid-movements-of-a-square example above is an example of a non-abelian group. All groups with 5 or fewer elements are Abelian.

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