Let A = {1, 2, 3}
The elements of $S_{3}$ are:-
$\{ p_{0} = 𝐼, p_{1} = (12), p_{2} = (13), p_{3} = (23), p_{4} = (123), p_{5} = (132)\}$
ORDERS:
1. The order of each of p₁, p₂, p₃, is 2.
2. The order of each of p₄, p₅, is 3.
INVERSES:
1. The inverse of the elements $p_{0}, p_{1}, p_{2}, p_{3}$ is the element itself.
2. $p_{4}^{-1} = p_{5}, p_{5}^{-1} = p_{4}$.
NON-ABELIAN:
As $p_{1}p_{2}= (132)$ and $p_{2}p_{1} =(123)$
Implies $p_{1}p_{2}\neq p_{2}p_{1}$ see more
Let S = {1, 2, 3, 4}
The elements of S4 are:-
{𝑝₀ = 𝐼,𝑝₁ = (12),𝑝₂ = (13),𝑝₃ = (14),𝑝₄ = (23),𝑝₅ = (24),𝑝₆ = (34),
𝑟₀ = (12)(34),𝑟₁ = (13)(24),𝑟₂ = (14)(23),
𝑎₀ = (123),𝑎₁ = (124),𝑎₂ = (132),𝑎₃ = (134),𝑎₄ = (142),a₅ = (143), 𝑎₆ = (234),𝑎₇ = (243),𝑏₀ = (1234),
𝑏₁ = (1243),𝑏₂ = (1324),𝑏₃ = (1342),𝑏₄ = (1423),𝑏₅ = (1432)
ORDERS:
1. The order of each of p₁, . . ., p₆ and r₀, . . ., r₃ is 2.
2. The order of each of a₀, . . ., a₇ is 3.
3. The order of each of b₀, . . ., b₅ is 4.
INVERSES:
1. The inverse of the elements 𝑝₀,p₁, . . . ,p₆ and r₀, . . ., r₃ is the element itself.
2. $𝑎_{0}^{-1}=𝑎_{2} , 𝑎_{1}^{-1}= 𝑎_{4} , 𝑎_{3}^{-1}= 𝑎_{5} , 𝑎_{6}^{-1}= 𝑎_{7}$ see more
The elements of S5 are:-
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$\{p_{0}=(1),p_{1}=(12),p_{2}=(13),p_{3}=(14),p_{4}=(15),p_{5}=(23),\\ p_{6}=(24),p_{7} =(25),p_{8}=(34),p_{9}=(35),p_{10}=(45)$,
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$a_{1}=(12)(34),a_{2}=(12)(35),a_{3}=(12)(45),a_{4}=(13)(24),a_{5}=(13)(25),\\
a_{6}=(13)(45),a_{7}=(14)(23),a_{8}=(14)(25),a_{9}=(14)(35), a_{10}=(15)(23),\\ a_{11}=(15)(24),a_{12}=(15)(34),a_{13}=(23)(45),a_{14}=(24)(35), a_{15}=(25)(34)$, |
$b_{1}=(123),b_{2}=(124),b_{3}=(125),b_{4}=(132),b_{5}=(134),\\ b_{6}=(135),b_{7}=(142),b_{8}=(143),b_{9}=(145),b_{10}=(152),\\
b_{11}=(153),b_{12}=(154),b_{13}=(234),b_{14}=(235), b_{15}=(243),\\b_{16}=(245),b_{17}=(253),b_{18}=(254), b_{19}=(345),b_{20}=(354)$, |
$c_{1}=(12)(345),c_{2}=(12)(354),c_{3}=(13)(245),c_{4}=(13)(254),c_{5}=(14)(235), \\
c_{6}=(14)(253), c_{7}=(15)(234), c_{8}=(15)(243), c_{9}=(23)(145), c_{10}=(23)(154),\\ c_{11}=(24)(135),c_{12}=(24)(153), c_{13}=(25)(134), c_{14}=(25)(143),c_{15}=(34)(125),\\ c_{16}=(34)(152),c_{17}=(35)(124), c_{18}=(35)(142), c_{19}=(45)(123), c_{20}=(45)(132)$, |
$r_{1}=(1234),r_{2}=(1243),r_{3}=(1235),r_{4}=(1253),r_{5}=(1245),\\ r_{6}=(1254),r_{7}=(1324),r_{8}=(1342),r_{9}=(1325),r_{10}=(1352),\\
r_{11}=(1345), r_{12}=(1354),r_{13}=(1423), r_{14}=(1432),r_{15}=(1425), \\ r_{16}=(1452), r_{17}=(1435), r_{18}=(1453),r_{19}=(1523), r_{20}=(1532),\\ r_{21}=(1524),r_{22}=(1542), r_{23}=(1534), r_{24}=(1543), r_{25}=(2345),\\ r_{26}=(2354), r_{27}=(2435), r_{28}=(2453),r_{29}=(2534), r_{30}=(2543)$, |
$f_{1}=(12345),f_{2}=(12435), f_{3}=(12354),f_{4}=(12534),f_{5}=(12453),\\
f_{6}=(12543),f_{7}=(13245), f_{8}=(13425), f_{9}=(13254), f_{10}=(13524),\\ f_{11}=(13452), f_{12}=(13542),f_{13}=(14235),f_{14}=(14325), f_{15}=(14253),\\ f_{16}=(14523),f_{17}=(14352), f_{18}=(14532),f_{19}=(15234),f_{20}=(15324),\\ f_{21}=(15243), f_{22}=(15423), f_{23}=(15342), f_{24}=(15432)\}$ |
1. The order of each of $p_{1}, .\: .\: .\:, p_{10}$ and $a_{1}, .\: .\: .\:, a_{15}$ is 2.
2. The order of each of $b_{1}, .\: .\: .\:, b_{20}$ is 3.
3. The order of each of $c_{1}, .\: .\: .\:, c_{20}$ is 6.
4. The order of each of $r_{1}, .\: .\: .\:, r_{30}$ is 4.
5. The order of each of $f_{1}, .\: .\: .\:, f_{24}$ is 5.
INVERSES:
1 1. The inverse of the elements 𝑝₀, p₁, . . . , p10 and a₀, . . ., a15 is the element itself.
2. $b_{1}^{-1}=b_{4},\quad b_{2}^{-1}=b_{7},\quad b_{3}^{-1}=b_{10},\quad b_{5}^{-1}=b_{8},\quad b_{6}^{-1}=b_{11}$,
$b_{9}^{-1}=b_{12},\quad b_{13}^{-1}=b_{15},\quad b_{14}^{-1}=b_{17},\quad b_{16}^{-1}=b_{18},\quad b_{19}^{-1}=b_{20}$ see more
2. $b_{1}^{-1}=b_{4},\quad b_{2}^{-1}=b_{7},\quad b_{3}^{-1}=b_{10},\quad b_{5}^{-1}=b_{8},\quad b_{6}^{-1}=b_{11}$,
$b_{9}^{-1}=b_{12},\quad b_{13}^{-1}=b_{15},\quad b_{14}^{-1}=b_{17},\quad b_{16}^{-1}=b_{18},\quad b_{19}^{-1}=b_{20}$ see more