# Nonstandard definitions of group

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Important terms have multiple definitions, from different perspectives. Some of these definitions describe the objects as special cases of more complicated objects.

By a definition of group we'll loosely mean something that gives rise to groups, and yields the same notion of isomorphism.

For historical definitions, refer historical definitions of group.

## As automorphisms of a structure

### Subgroup of a symmetric group

Naive definition: A group is a collection of permutations on a set, that contains the identity permutation, is closed under inversion, and is closed under composition. In other words, a group is a subgroup of a symmetric group.

Justification for this definition: Cayley's theorem, which states that every group can be embedded as a subgroup of a symmetric group, via the regular action: the action on itself by left multiplication.

Problem with this definition: The chief problem with this definition is that it doesn't recognize that the same group could arise in totally different ways as a subgroup of a symmetric group. For instance, in the symmetric group on four elements, the Klein-four group occurs in two distinct ways: one as a normal subgroup including the double transpositions, and the other, as a direct product of 2-element subgroups generated by disjoint transpositions (e.g., $\langle (1 2), (3 4) \rangle$).

Modification to the definition: The definition can be rectified by including what it means for two groups to be isomorphic.

### Subgroup of a general linear group

Naive definition: A group is a collection of invertible linear transformation on a vector space, that contains the identity transformation, and is closed under composition and inversion of transformations. In other words, a group is a subgroup of a general linear group.

Justification for this definition: Every group can be embedded in a symmetric group, and the symmetric group on a set embeds into the vector space with that set as basis (note: the set may be infinite, so the vector space may be infinite-dimensional).

Problem with this definition: As in the case of the previous definition, the definition fails to capture the notion of isomorphism of groups.

Modification to the definition: The definition can be rectified by including what it means for two groups to be isomorphic.

### Category with one object

Definition: A group is a (small) category with one object, where every morphism is invertible. Equivalently a group is a groupoid with one object. Two groups are isomorphic if the corresponding categories are equivalent as categories (or equivalently, are isomorphic as categories).

Justification for this definition: A category with one object is completely determined by the monoid of morphisms from that object to itself. When every morphism is invertible, this monoid is a group ,and the category is completely determined by that group. Conversely, given any group, we can set up a category with one object, where the automorphism group of that object is the given group.

## Group with (absent) additional structure

### Discrete topological group

Definition: A group is a discrete topological group.

Justification for this definition: We can embed the category of groups inside the category of topological groups by sending each group to the corresponding discrete topological group. In fact, this embedding is the left-adjoint functor to the forgetful functor from topological groups to groups, that sends a topological group to its underlying group. The notion of discrete topological group also resonates, for instance, with the general idea that covering spaces and fundamental groups are a special case of principal bundles for a topological group.

### Group object in the category of sets

Definition: A group is a group object in the category of sets (here, the monoidal product is Cartesian product).

Justification for the definition: The notion of group object was invented to generalize the idea of groups to structures that are sets with additional structure, or are structures without an underlying set. When we specialize to the category of sets, we recover the original notion of group.

### Hopf algebra in the category of sets

Definition: A group is a Hopf algebra in the category of sets.

## As strengthenings of its weakenings

Also see: Varying group

1. A group is a monoid where every element has a two-sided inverse
2. A group is an associative algebra loop, or equivalently, is an associative quasigroup