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 -

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

- The set of integers with the operation of addition.
- The set of positive rational numbers with the operation of multiplication.
- The set with the single element
**1**and the operation * defined so that**1*****1**=**1**(The*trivial group*) - The set of all rigid movements of a square:
- Flip the square vertically
- Flop the square horizontally
- Flip the square across the northeast-southwest diagonal
- Flip the square across the northwest-southeast diagonal
- Rotate the square 90 degrees clockwise
- Rotate the square 180 degrees
- Rotate the square 90 degrees counterclockwise
- 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.)

- The set of positive integers with the operation of subtraction. (The operation is not closed, because 7-12 is not a positive integer.)
- 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.)
- The set of positive integers with the operation of addition. (There is no identity element)
- The set of integers with the operation of multiplication. (The identity is 1, but the element 0 has no inverse.)

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