If-C-r-C-s-are-cyclic-groups-such-that-g-c-d-r-s-1-then-show-that-C-r-C-s-is-a-cyclic-group-Mastermind-

Question Number 171314 by Mastermind last updated on 12/Jun/22

I f C_{r}, C_{s} a r e c y c l i c g r o u p s s u c h t h a t

g . c . d (r, s) = 1, t h e n s h o w t h a t C_{r} \times C_{s} i s

a c y c l i c g r o u p .

M a s t e r m i n d

Answered by mindispower last updated on 12/Jun/22

C_{r} c y c l i c \Rightarrow \exists a \in C_{r} \dots s u c h \forall g \in C_{r} \exists n \in {0, \dots . r - 1}

a^{n} = g

C_{s} c y c l i c \Rightarrow \exists b \in C_{s} . . s u c h \forall g^{'} \in C_{s} . \exists m \in {0, \dots . s - 1}

b^{m} = g^{'}

w e u s e s t a n d a r d e f i n i t i o n o f p r o d u c t o f / G r o u p s

c a r d (C_{r} * C_{s}) ⩽ r s

(a, b) * \dots . . (a, b) \dots (r s) = ({(a^{r})}^{s}, {(b^{s})}^{r}) = (e, e^{'}) T i m e s

(a^{x}, b^{x}) = (e, e^{'}) \Rightarrow r ∣ x, s ∣ x \Rightarrow r s ∣ x \dots g c d (r, s) = 1

\Rightarrow r s ∣ c a r d (C_{r} * C_{s}) \Rightarrow c a r d (C_{r} * C_{s}) = r s

a n d C_{r} * C_{s} =< (a, b) >

Commented by Mastermind last updated on 14/Jun/22

T h a n k s

Commented by mindispower last updated on 19/Jun/22

w i t h e p l e a s u r