{{{id=1|
G = SymmetricGroup(5)
///
}}}

{{{id=31|
G.order()
///
120
}}}

{{{id=2|
G.list()(1,2)(4,5)
///
[(), (4,5), (3,4), (3,4,5), (3,5,4), (3,5), (2,3), (2,3)(4,5), (2,3,4), (2,3,4,5), (2,3,5,4), (2,3,5), (2,4,3), (2,4,5,3), (2,4), (2,4,5), (2,4)(3,5), (2,4,3,5), (2,5,4,3), (2,5,3), (2,5,4), (2,5), (2,5,3,4), (2,5)(3,4), (1,2), (1,2)(4,5), (1,2)(3,4), (1,2)(3,4,5), (1,2)(3,5,4), (1,2)(3,5), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5), (1,2,3,5,4), (1,2,3,5), (1,2,4,3), (1,2,4,5,3), (1,2,4), (1,2,4,5), (1,2,4)(3,5), (1,2,4,3,5), (1,2,5,4,3), (1,2,5,3), (1,2,5,4), (1,2,5), (1,2,5,3,4), (1,2,5)(3,4), (1,3,2), (1,3,2)(4,5), (1,3,4,2), (1,3,4,5,2), (1,3,5,4,2), (1,3,5,2), (1,3), (1,3)(4,5), (1,3,4), (1,3,4,5), (1,3,5,4), (1,3,5), (1,3)(2,4), (1,3)(2,4,5), (1,3,2,4), (1,3,2,4,5), (1,3,5,2,4), (1,3,5)(2,4), (1,3)(2,5,4), (1,3)(2,5), (1,3,2,5,4), (1,3,2,5), (1,3,4)(2,5), (1,3,4,2,5), (1,4,3,2), (1,4,5,3,2), (1,4,2), (1,4,5,2), (1,4,2)(3,5), (1,4,3,5,2), (1,4,3), (1,4,5,3), (1,4), (1,4,5), (1,4)(3,5), (1,4,3,5), (1,4,2,3), (1,4,5,2,3), (1,4)(2,3), (1,4,5)(2,3), (1,4)(2,3,5), (1,4,2,3,5), (1,4,2,5,3), (1,4,3)(2,5), (1,4)(2,5,3), (1,4,3,2,5), (1,4)(2,5), (1,4,2,5), (1,5,4,3,2), (1,5,3,2), (1,5,4,2), (1,5,2), (1,5,3,4,2), (1,5,2)(3,4), (1,5,4,3), (1,5,3), (1,5,4), (1,5), (1,5,3,4), (1,5)(3,4), (1,5,4,2,3), (1,5,2,3), (1,5,4)(2,3), (1,5)(2,3), (1,5,2,3,4), (1,5)(2,3,4), (1,5,3)(2,4), (1,5,2,4,3), (1,5,3,2,4), (1,5)(2,4,3), (1,5,2,4), (1,5)(2,4)]
}}}

{{{id=3|
G.is_abelian()
///
False
}}}

{{{id=4|
G.is_cyclic()
///
False
}}}

{{{id=5|
a = G([(1,2), (3, 4, 5)])
///
}}}

{{{id=34|
a*a
///
(3,5,4)
}}}

{{{id=33|
a*a*a
///
(1,2)
}}}

{{{id=8|
a^6
///
()
}}}

{{{id=6|
a.order()
///
6
}}}

{{{id=7|
[b.order() for b in G]
///
[1, 2, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 3, 4, 2, 3, 2, 4, 4, 3, 3, 2, 4, 2, 2, 2, 2, 6, 6, 2, 3, 6, 4, 5, 5, 4, 4, 5, 3, 4, 6, 5, 5, 4, 4, 3, 5, 6, 3, 6, 4, 5, 5, 4, 2, 2, 3, 4, 4, 3, 2, 6, 4, 5, 5, 6, 6, 2, 5, 4, 6, 5, 4, 5, 3, 4, 6, 5, 3, 4, 2, 3, 2, 4, 4, 5, 2, 6, 6, 5, 5, 6, 6, 5, 2, 4, 5, 4, 4, 3, 5, 6, 4, 3, 3, 2, 4, 2, 5, 4, 6, 2, 5, 6, 6, 5, 5, 6, 4, 2]
}}}

{{{id=11|
G.order()
///
120
}}}

{{{id=17|
G = DihedralGroup(10)
///
}}}

{{{id=18|
G.order()
///
20
}}}

{{{id=12|
H = G.center(); H
///
Subgroup of (Dihedral group of order 20 as a permutation group) generated by [(1,6)(2,7)(3,8)(4,9)(5,10)]
}}}

{{{id=19|

///
}}}

{{{id=9|
H.is_normal(G)
///
True
}}}

{{{id=10|
H.is_abelian()
///
True
}}}

{{{id=13|
K = G.quotient(H)
///
}}}

{{{id=14|
K.order()
///
10
}}}

{{{id=15|
G.order()/H.order()  ## teorema di Lagrange
///
10
}}}

{{{id=16|
H.order()
///
2
}}}

{{{id=20|
a = H[1]
///
}}}

{{{id=35|
H
///
Subgroup of (Dihedral group of order 20 as a permutation group) generated by [(1,6)(2,7)(3,8)(4,9)(5,10)]
}}}

{{{id=36|
a^2
///
()
}}}

{{{id=21|
## verifico che H e' il centro di G, cioe' che ogni a in H permuta con tutti gli elementi di G
///
}}}

{{{id=22|
for g in G:
    a*g == g*a
///
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
}}}

{{{id=23|
G = SymmetricGroup(5)
///
}}}

{{{id=24|
G.is_abelian()
///
False
}}}

{{{id=25|
a = G.random_element()
///
}}}

{{{id=26|
H = G.subgroup([a])
///
}}}

{{{id=27|
H.order()
///
5
}}}

{{{id=28|
H.is_abelian()
///
True
}}}

{{{id=29|
H.is_cyclic()
///
True
}}}

{{{id=30|
gap(factorial(300))
///
306057512216440636035370461297268629388588804173576999416776741259476533176716867465515291422477573349939147888701726368864263907759003154226842927906974559841225476930271954604008012215776252176854255965356903506788725264321896264299365204576448830388909753943489625436053225980776521270822437639449120128678675368305712293681943649956460498166450227716500185176546469340112226034729724066333258583506870150169794168850353752137554910289126407157154830282284937952636580145235233156936482233436799254594095276820608062232812387383880817049600000000000000000000000000000000000000000000000000000000000000000000000000
}}}

{{{id=37|

///
}}}