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Question Number 111458 by mohammad17 last updated on 03/Sep/20

Commented by kaivan.ahmadi last updated on 03/Sep/20

$Q_{1}$ ${(a b)}^{n} = (a b) (a b) . . . (a b); n t i m e s$ $a (b a) (b a) . . . . (b a) b = a (a b) (a b) . . . (a b) b =$ $a^{2} (b a) . . . . (b a) b^{2}$ $c o n t i n u e t h i s w a y w e g e t r e s u l t .$
Commented by kaivan.ahmadi last updated on 04/Sep/20

$Q_{5}$ $l e t O (G) = 2 n$ $f o r e a c h e \neq x \in G \Rightarrow \exists n : x^{2 n} = e \Rightarrow {(x^{n})}^{2} = e$ $b u t x^{n} \in G .$ $s a y x^{n} = a s o a^{2} = e .$ $[x^{n} \neq x o t h e r w i s e i f x^{n} = x \Rightarrow x^{n - 1} = e \Rightarrow$ $o (x) = n - 1 t h a t i s a c o n t r u d i c t i o n]$
Commented by kaivan.ahmadi last updated on 03/Sep/20

$Q_{4}$ $l e t G h a s m e l e m e n t x_{i}; i = 1, 2, . ., m$ $l e t d_{i} = O (x_{i}) f o r e a c h x_{i} \in G$ $n = l c m (d_{i}) .$
Commented by kaivan.ahmadi last updated on 04/Sep/20
$Q_2 S_3 ={e,(1 2),(1 3),(2 3),(1 2 3),(1 3 2)} the elements e,(1 2),(1 3),(2 3) have order 2 and the elements e,(1 2 3),(1 3 2) have order 3 3$
$Q_{2}$ $S_{3} = {e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2)}$ $t h e e l e m e n t s e, (1 2), (1 3), (2 3) h a v e o r d e r 2$ $a n d t h e e l e m e n t s e, (1 2 3), (1 3 2) h a v e o r d e r 3$ $3$
Commented by kaivan.ahmadi last updated on 04/Sep/20
$Q_3 let G={e,a,b} ab≠a sice if ab=a⇒b=e similarly ab≠b so ab=e similarly ba=e and so ab=ba then G is abelian. && the group of order p^2 where p is prime is an abelian group so if O(G)=4=2^2 then G is abelian$
$Q_{3}$ $l e t G = {e, a, b}$ $a b \neq a s i c e i f a b = a \Rightarrow b = e$ $s i m i l a r l y a b \neq b$ $s o a b = e$ $s i m i l a r l y b a = e a n d s o a b = b a t h e n G i s$ $a b e l i a n .$ $& &$ $t h e g r o u p o f o r d e r p^{2} w h e r e p i s p r i m e i s$ $a n a b e l i a n g r o u p$ $s o i f O (G) = 4 = 2^{2} t h e n G i s a b e l i a n$
Commented by kaivan.ahmadi last updated on 04/Sep/20

$$ $$ $e v e r y g r o u p o f o r d e r 5 i s a b e l i a n . y o u c a n$ $f i n d t h i s t h e o r e m i n e l e m e n t a r y a l g e b r a$ $b o o k s .$ $& &$ $g e n e r a l l y e v e r y g o r u p o f o r d e r l e s s t h a n$ $6 i s a b e l i a n .$
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