# Category:Symmetric Groups

This category contains results about Symmetric Groups.

Definitions specific to this category can be found in Definitions/Symmetric Groups.

Let $S$ be a set.

Let $\map \Gamma S$ denote the set of permutations on $S$.

Let $\struct {\map \Gamma S, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings.

Then $\struct {\map \Gamma S, \circ}$ is called the **symmetric group on $S$**.

If $S$ has $n$ elements, then $\struct {\map \Gamma S, \circ}$ is often denoted $S_n$.

## Subcategories

This category has the following 13 subcategories, out of 13 total.

### A

### C

### E

### G

### P

### S

## Pages in category "Symmetric Groups"

The following 55 pages are in this category, out of 55 total.

### A

### C

### E

### G

### N

### P

- Parity Function is Homomorphism
- Parity of Conjugate of Permutation
- Parity of Inverse of Permutation
- Parity of K-Cycle
- Permutation Group is Subgroup of Symmetric Group
- Permutation Induces Equivalence Relation
- Permutation Induces Equivalence Relation/Corollary
- Permutation of Cosets
- Permutation of Cosets/Corollary 1
- Permutation of Cosets/Corollary 2
- Power of Moved Element is Moved
- Powers of Disjoint Permutations
- Powers of Permutation Element

### S

- Set of Invertible Mappings forms Symmetric Group
- Sign of Permutation is Plus or Minus Unity
- Stabilizer in Group of Transformations
- Subgroup of Symmetric Group that Fixes n
- Subgroups of Symmetric Group Isomorphic to Product of Subgroups
- Symmetric Group has Non-Normal Subgroup
- Symmetric Group is Group
- Symmetric Group is not Abelian
- Symmetric Group is Subgroup of Monoid of Self-Maps
- Symmetric Group on 3 Letters
- Symmetric Group on 4 Letters
- Symmetric Group on n Letters is Isomorphic to Symmetric Group
- Symmetric Groups of Same Order are Isomorphic