# Category:Symmetric Groups

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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 17 subcategories, out of 17 total.

### A

### C

### E

### G

### P

### S

- Sign of Permutation (3 P)
- Symmetric Group is Group (3 P)

### T

- Transitive Subgroups (3 P)

## Pages in category "Symmetric Groups"

The following 63 pages are in this category, out of 63 total.

### A

### C

- Cayley's Representation Theorem
- Center of Symmetric Group is Trivial
- Composition Series/Examples/Symmetric Group Sn for n gt 2 where n ne 4
- Conjugacy Classes of Symmetric Group
- Conjugates of Transpositions
- Conjugation of Bijection between Symmetric Groups is Isomorphism
- Cycle Decomposition of Conjugate

### E

### G

### I

### N

### P

- Parity Function is Homomorphism
- Parity of Conjugate of Permutation
- Parity of Inverse of Permutation
- 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
- Set of Transpositions is not Subgroup of 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 Generated by Transposition and n-Cycle
- Symmetric Group is Group
- Symmetric Group is not Abelian
- Symmetric Group is Subgroup of Monoid of Self-Maps
- Symmetric Group on Greater than 4 Letters is Not Solvable
- Symmetric Group on n Letters is Isomorphic to Symmetric Group
- Symmetric Groups of Same Order are Isomorphic