The element that is its own inverse

The element that is the same as every other element

The element that is the only element of the group

The element that is the first element of the group

The number of elements in the group

The number of times the element appears in the group

The smallest positive integer $n$ such that $a^n = e$

The largest positive integer $n$ such that $a^n = e$

A set of elements that is closed under the group operation

A set of elements that contains the identity element

A set of elements that is non-empty

A set of elements that is finite

A subgroup that is invariant under conjugation

A subgroup that is closed under the group operation

A subgroup that contains the identity element

A subgroup that is non-empty

The group obtained by dividing the group by the normal subgroup

The group obtained by taking the union of the group and the normal subgroup

The group obtained by taking the intersection of the group and the normal subgroup

The group obtained by taking the complement of the normal subgroup in the group

The set of all elements that commute with every other element in the group

The set of all elements that are their own inverses

The set of all elements that are in the center of the group

The set of all elements that are in the center of the group

The subgroup generated by all the commutators of the group

The subgroup generated by all the elements of the group

The subgroup generated by all the inverses of the elements of the group

The subgroup generated by all the squares of the elements of the group

The subgroup generated by all the commutators of the group

The subgroup generated by all the elements of the group

The subgroup generated by all the inverses of the elements of the group

The subgroup generated by all the squares of the elements of the group

The group generated by the set of generators subject to no relations

The group generated by the set of generators subject to the relation that every generator is its own inverse

The group generated by the set of generators subject to the relation that every generator commutes with every other generator

The group generated by the set of generators subject to the relation that every generator is equal to the identity element

The group of all homotopy classes of loops in the space

The group of all homotopy classes of paths in the space

The group of all homotopy classes of spheres in the space

The group of all homotopy classes of disks in the space

The group of all homology classes of cycles in the space

The group of all homology classes of boundaries in the space

The group of all homology classes of spheres in the space

The group of all homology classes of disks in the space

The group of all cohomology classes of cocycles in the space

The group of all cohomology classes of coboundaries in the space

The group of all cohomology classes of spheres in the space

The group of all cohomology classes of disks in the space

The group of all K-theory classes of vector bundles in the space

The group of all K-theory classes of sphere bundles in the space

The group of all K-theory classes of disk bundles in the space

The group of all K-theory classes of line bundles in the space

The group of all L-theory classes of generalized homology theories in the space

The group of all L-theory classes of cohomology theories in the space

The group of all L-theory classes of K-theory theories in the space

The group of all L-theory classes of stable homotopy theories in the space

The group of all motivic cohomology classes of algebraic varieties in the space

The group of all motivic cohomology classes of schemes in the space

The group of all motivic cohomology classes of stacks in the space

The group of all motivic cohomology classes of categories in the space