Changes

Symmetric group:S5

, 00:45, 19 August 2012
Arithmetic functions
| {{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]]): $\! r(q^3 - q) = 2(4^3 - 4) = 2(60) = 120$.
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| {{arithmetic function value exponent given order|60|120}} || As $\! S_kS_n, k n = 5:$ $\! \operatorname{lcm} \{ 1,2,\dots,n \} = \operatorname{lcm} \{1,2,3,4,5 \} = 60$ <br><br> As $\! PGL(2,q), q = p = 5:$ (where $\! p$ is the underlying prime for $\! q$) $\! p(q^2 - 1)/2 = 5 \cdot (5^2 - 1)/2 = 60$
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| [[derived length]] || -- || || not a solvable group.