Ques. 11 Let {H_α } ∈ Ω be a family of subgroup of a group G then prove that ∩_(α=Ω) H_α is also a subgroup Ques. 12 Using GAP, find the elements A, B and C in D


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Question Number 193893 by Mastermind last updated on 22/Jun/23
$Ques. 11 Let {H_α } ∈ Ω be a family of subgroup of a group G then prove that ∩_(α=Ω) H_α is also a subgroup Ques. 12 Using GAP, find the elements A, B and C in D_5 such that AB = BC but A ≠ C.$
$Ques . 11$ $Let {H_{α}} \in Ω be a family of subgroup of$ $a group G then prove that \underset{α = Ω}{\cap} H_{α} is also a$ $subgroup$ $Ques . 12$ $Using GAP, find the elements A, B and$ $C in D_{5} such that AB = BC but A \neq C .$
	Answered by Rajpurohith last updated on 22/Jun/23

	$(11) L e t U = \cap_{α} (H_{α})$ $w e n e e d t o p r o v e t h a t U i s s u b g r o u p o f G .$ $A s e \in H_{α} \forall α,$ $e \in \cap_{α} H_{α}$ $\Rightarrow U \neq \emptyset .$ $l e t a, b \in U$ $\Rightarrow a, b \in H_{α} \forall α$ $s i n c e H_{α} i s a s u b g r o u p f o r e a c h α,$ ${a b}^{- 1} \in H_{α} \forall α$ $\Rightarrow {a b}^{- 1} \in (\cap H_{α}) \forall α$ $\Rightarrow {a b}^{- 1} \in U \forall a, b \in U$ $\Rightarrow U = \cap_{α} H_{α} i s a s u b g r o u p o f G .$
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