Let G=S_{n}\ \text{and let}\ H=\{\alpha \in G|\alpha(1)=1\}. Show that H is

Caitlin Esparza

Caitlin Esparza

Answered question

2022-02-13

Let G=Sn and let H={α∈G∣α(1)=1}.
Show that H is a subgroup of G.

Answer & Explanation

prudomiajpt

prudomiajpt

Beginner2022-02-14Added 15 answers

(H,∘) is a subgroup of (G,∘) iff
1) H≠ϕ
2) α,β∈H⇒α⋅β∈H
3) α∈H⇒α−1∈H
G=Sn H={α∈Sn∣α(1)=1}
identify permutation e∈H, so H≠ϕ.
Let α,β∈H then α(1)=1 and β(1)=1
⇒α⋅β(1)=α(β(1))=α(1)=1
⇒(αβ)(i)=1
⇒αβ∈H
Let α∈H⇒α(1)=1⇒α−1(1)=1⇒α−1∈H
Thus, H is a subgroup of G=Sn

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