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## Symmetric group:S5

, 16:24, 23 July 2013
Basic properties
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|120}} || As $\! S_n, n = 5:$ $\! n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ <br><br> As $\! PGL(2,q), q = 5$ (see [[order formulas for linear groups of degree two]]): $\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(4)(6) = 120$<br>As $\! P\Gamma L(2,q), q = p^r, q = 4, p = 2, r = 2$ (see [[order formulas for linear groups of degree two]]): $\! r(q^3 - q) = 2(4^3 - 4) = 2(60) = 120$.<br><br>See [[element structure of symmetric group:S5#Order computation]] for more information.
|-
| {{arithmetic function value exponent given order|60|120}} || As $\! S_n, n = 5:$ $\! \operatorname{lcm} \{ 1,2,\dots,n \} = \operatorname{lcm} \{1,2,3,4,5 \} = 60$ <br><br> As $\! PGL(2,q), q = p = 5:$ (where $\! p$ is the underlying prime for $\! q$) $\! p(q^2 - 1)/2 = 5 \cdot (5^2 - 1)/2 = 60$
===Basic properties===
{| class="sortable" border="1"
!Property !! Meaning !! Satisfied? !! Explanation !! Comment
|-
|[[Dissatisfies property::abelian group]] || any two elements commute || No || $(1,2)$, $(1,3)$ don't commute || $S_n$ is non-abelian, $n \ge 3$.
|-
|[[Dissatisfies property::nilpotent group]] || has a [[central series]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || $S_n$ is non-nilpotent, $n \ge 3$.
|-
|[[Dissatisfies property::metacyclic group]] || has a [[cyclic normal subgroup]] with a cyclic [[quotient group]] || No || No [[cyclic normal subgroup]] || $S_n$ is not metacyclic, $n \ge 4$.
|-
|[[Dissatisfies property::supersolvable group]] || has a [[normal series]] of finite length with all successive quotients cyclic groups. || No || No [[cyclic normal subgroup]] || $S_n$ is not supersolvable, $n \ge 4$.
|-
|[[Dissatisfies property::solvable group]] || has a [[normal series]] of finite length with all successive quotients abelian groups. || No || The subgroup $A_5$ [[A5 is simple|is simple non-abelian ]] || $A_n$ is simple and hence $S_n$ not solvable, $n \ge 5$.
|-
|[[Dissatisfies property::simple non-abelian group]] || has no proper nontrivial [[normal subgroup]]. || No || has a proper nontrivial normal subgroup [[A5 in S5]]. ||
|-
|[[Satisfies property::almost simple group]] || contains a centralizer-free [[simple normal subgroup]]. || Yes || It contains a centralizer-free [[simple normal subgroup]], namely [[A5 in S5]]. || [[symmetric groups are almost simple]] for degree 5 or higher.
|-
|[[Dissatisfies property::perfect group]] || equals its own [[derived subgroup]] || No || Its derived subgroup is [[A5 in S5]] and abelianization is [[cyclic group:Z2]]. ||
|-
|[[Dissatisfies property::quasisimple group]] || perfect, and [[inner automorphism group]] is simple non-abelian. || No || Follows from not being perfect. ||
|-
|[[Satisfies property::centerless group]] || [[center]] is trivial. || Yes || [[symmetric groups are centerless]] || [[symmetric groups are centerless]] for degree other than two.
|-
|[[Satisfies property::complete group]] || centerless, and every automorphism is inner. || Yes || Centerless and every automorphism's inner || [[Symmetric groups are complete]] except the ones of degree $2,6$.
|}
|-
|[[Satisfies property::one-headed group]] || Yes || The alternating group is the unique [[maximal normal subgroup]] || True for all $n > 1$.
|-
|[[satisfies property::N-group]] || Yes || See [[classification of symmetric groups that are N-groups]] || True only for $n \le 6$.
|}
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