All elements of the set must have an ____ under the given operation.

A ring that has multiplicative inverses and thus allows for division and is communative under multiplication. Include the rational numbers, real numbers, and complex numbers.

A special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them

Any group that simply rearranges the element of a group into a distinct order.

Any group that can be generated by repeatedly applying the operation to a single element of the group.

Any group in which the commutative property applies.

A homomorphism that is bijective. This means that 2 groups have an identical size and structure related to their 2 operations.

All groups must have this property. a+(b+c)=(a+b)+c

Hint

Answer

An operation is considered close on a set when it always results in an answer that is an element of the original set.

An abelian group with a second operation that has the other group properties. In addition, the distributive property must apply between the two operations.

A transformation of one set into another that preserves in the second set the relations between elements of the first.

The group of all permutations of a set.

Number of elements in a group.

The smallest non-cyclic group. Often symbolized by letter V. Has only four elements, and ever four-element non-cyclic group is isomorphic to it.

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