In-the-symmetric-group-S-n-o-let-H-denotes-the-set-of-permutations-leaving-the-integer-n-fixed-H-f-S-n-f-n-n-Show-that-the-pair-H-o-is-subgroup-of-S-n-o-note-the-operatio

Question Number 76303 by arkanmath7@gmail.com last updated on 26/Dec/19

I n t h e s y m m e t r i c g r o u p (S_{n}, o), l e t

H d e n o t e s t h e s e t o f p e r m u t a t i o n s

l e a v i n g t h e i n t e g e r n f i x e d :

H = {f \in S_{n} ∣ f (n) = n}

S h o w t h a t t h e p a i r (H, o) i s s u b g r o u p

o f (S_{n}, o) .

^{'} n o t e : t h e o p e r a t i o n i s {c o m p o s i t i o n}^{'}

Answered by mind is power last updated on 26/Dec/19

Id \in H

let f, g \in H

fog \in H since

fog (n) = f (g (n)) = f (n) = n

if f \in H f^{-} \in H

since f \in S_{n} \Rightarrow f is bijection \Rightarrow f^{-} existe un

f (n) = n \Rightarrow f^{-} of (n) = f^{-} (n) \Rightarrow n = f^{-} (n)

\Rightarrow H is a sub groupe of (S_{n_{,}}, o)

Commented by arkanmath7@gmail.com last updated on 26/Dec/19

T h a n k s