Ques. 1 Let G be a group and b a fixed element of G. Prove that the map G into G given by x→bx is bijective Ques. 2 Let G be a group and g be an element of G. Prove that a) (g^(−1) )^(−1) =g b) g^m g^n = g^(m+n) Help!


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Question Number 193711 by Mastermind last updated on 18/Jun/23

$Ques . 1$ $Let G be a group and b a fixed element$ $of G . Prove that the map G into G given$ $by x \to bx is bijective$ $Ques . 2$ $Let G be a group and g be an element$ $of G . Prove that$ $a) {(g^{- 1})}^{- 1} = g$ $b) g^{m} g^{n} = g^{m + n}$ $Help!$
	Answered by Rajpurohith last updated on 19/Jun/23

	$(1) G b e a g r o u p, b \in G b e f i x e d .$ $d e f i n e f : G \to G b y f (x) = b x \forall x \in G$ $c l e a r l y f i s w e l l d e f i n e d .$ $s u p p o s e f (c) = f (d) f o r c, d \in G$ $\Rightarrow b c = b d, b y c a n c e l l a t i o n p r o p e r t y c = d .$ $s o f i s a n i n j e c t i v e f u n c t i o n f r o m G t o i t s e l f .$ $l e t y \in G s o f (b^{- 1} y) = {b b}^{- 1} y = y$ $H e n c e f i s s u r j e c t i v e .$ $T h u s f i s b i j e c t i v e . ◼$ $(2) (a) l e t G \in g s a y h = g^{- 1} \Rightarrow h^{- 1} = {(g^{- 1})}^{- 1}$ $\Rightarrow h g = e$ $\Rightarrow h^{- 1} (h g) = h^{- 1}$ $\Rightarrow (h^{- 1} h) g = h^{- 1}$ $\Rightarrow g = h^{- 1} = {(g^{- 1})}^{- 1} ∴ {(g^{- 1})}^{- 1} = g ◼$ $(b) g^{m} . g^{n} = {(g . g . g . . . g)}_{m t i m e s} {(g . g . g . . . g)}_{n t i m e s}$ $= {(g . g . g . . . g)}_{m + n t i m e s} = g^{m + n} ◼$
	Commented by Mastermind last updated on 20/Jun/23

	$Thank you so much$
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