Ques-11-Let-H-be-a-family-of-subgroup-of-a-group-G-then-prove-that-H-is-also-a-subgroup-Ques-12-Using-GAP-find-the-elements-A-B-and-C-in-D-5-such-that-AB-BC-but

Question Number 193893 by Mastermind last updated on 22/Jun/23

Ques . 11

Let {H_{α}} \in Ω be a family of subgroup of

a group G then prove that \underset{α = Ω}{\cap} H_{α} is also a

subgroup

Ques . 12

Using GAP, find the elements A, B and

C in D_{5} such that AB = BC but A \neq C .

Answered by Rajpurohith last updated on 22/Jun/23

(11) L e t U = \cap_{α} (H_{α})

w e n e e d t o p r o v e t h a t U i s s u b g r o u p o f G .

A s e \in H_{α} \forall α,

e \in \cap_{α} H_{α}

\Rightarrow U \neq \emptyset .

l e t a, b \in U

\Rightarrow a, b \in H_{α} \forall α

s i n c e H_{α} i s a s u b g r o u p f o r e a c h α,

{a b}^{- 1} \in H_{α} \forall α

\Rightarrow {a b}^{- 1} \in (\cap H_{α}) \forall α

\Rightarrow {a b}^{- 1} \in U \forall a, b \in U

\Rightarrow U = \cap_{α} H_{α} i s a s u b g r o u p o f G .