exam midterm 2

M I D T E R M 2 A N J I A N G P r o b l e m 1 . ( S e c t i o n 6 . 3 . 1 4 ) P r o o f . F o r t h e g r o u p G p , w e h a v e n p ≡ 1 mod p a n d n p | 3 , t h u s n p = 1 . T h e n < a > i s t h e o n l y S y l o w p - s u b g r o u p o f G p . N o t e e v e r y n o n i d e n t i t y e l e m e n t o f G p h a s o r d e r 3 o r p . I f | ab |= p t h e n < ab >=< a > , ab ∈< a > , i . e . , ab = a n+1 f o r s o m e n ∈ Z + , i . e . , b = a n ∈< a > , t h u s b p = 1 a n d t h e n 3 | p , a c o n t r a d i c t i o n . S o | ab |= 3 . S i m i l a r l y , i f | ab 2 |= p t h e n < ab 2 >=< a > a n d ab 2 ∈< a > , i . e . , ab 2 = a m+1 f o r s o m e m ∈ Z + , i . e . , b 2 = a m ∈< a > , t h u s b 2 p = 1 a n d t h e n 3 | 2 p , s o 3 | p , a c o n t r a d i c t i o n . S o | ab 2 |= 3 . T h u s G i s a s u b r i n g o f F (G p ) , s o t h e s u r j e c t i v e h o m o m o r p h i s m f r o m F (G p ) t o G p i s a l s o s u r j e c t i v e w h e n r e s t r i c t e d o n G , i . e . , G p i s a h o m o m o r p h i c i m a g e o f G . I f G i s a n i t e g r o u p , a n d s u p p o s e t h a t | G |= n ∈ Z + , w e c a n a l w a y s n d a p r i m e p w h i c h i s c o n g r u e n t t o 1 m o d 3 a n d n < 3 p . T h i s c o n t r a d i c t s w i t h t h e f a c t t h a t G p i s a h o m o m o r p h i c i m a g e o f G , t h u s | G p | should be less or equal to | G | . P r o b l e m 2 . ( S e c t i o n 7 . 1 . 3 0 ) P r o o f . ( a ) W e c o m p u t e ϕψ(a 1 , a 2 , a 3 ,...) = ϕ(0,a 1 ,a 2 ,a 3 ,...) = (a 1 , a 2 , a 3 ,...) , ψϕ(a 1 ,a 2 ,a 3 ,...) = ψ(a 2 ,a 3 ,...) = (0,a 2 ,a 3 ,...) . S o ϕψ = I d(∈ R) a n d ψϕ = I d . ( b ) L e t ψ c (a 1 ,a 2 ,a 3 ,...) = (c, a 1 ,a 2 ,a 3 ,...) w i t h a n y c o n s t a n t c , t h e n ψ c s a r e i n n i t e l y m a n y r i g h t i n v e r s e s f o r ϕ . ( c ) L e t π(a 1 , a 2 , a 3 ,...) = (a 1 , 0, 0,...) . ∀(a 1 1 , a 1 2 , a 1 3 ,...), (a 2 1 , a 2 2 , a 2 3 ,...) ∈ Z × Z × Z×··· , π[(a 1 1 , a 1 2 , a 1 3 ,...) + (a 2 1 ,a 2 2 ,a 2 3 ,...)] = π(a 1 1 + a 2 1 , a 1 2 + a 2 2 , a 1 3 + a 2 3 ,...) = a 1 1 + a 2 1 = π(a 1 1 , a 1 2 ,a 1 3 ,...) + π(a 2 1 , a 2 2 , a 2 3 ,...) . T h u s π ∈ R . C h e c k ϕπ(a 1 ,a 2 ,a 3 ,...) = ϕ(a 1 , 0, 0,...) = (0, 0, 0,...) , πϕ(a 1 ,a 2 ,a 3 ,...) = π(a 2 , a 3 ,...) = (a 2 , 0, 0,...) . S o ϕπ = 0 b u t πϕ = 0 . ( d ) A s s u m e t h a t t h e r e i s a n o n z e r o e l e m e n t λ ∈ R s u c h t h a t λϕ = 0 . T h e n λϕ(a 1 ,a 2 ,a 3 ,...) = λ(a 2 , a 3 ,...) = (0, 0, 0,...) , ∀(a 2 1 ,a 2 2 ,a 2 3 ,...) ∈ Z × Z × Z×··· . S o λ = 0 , a c o n t r a d i c t i o n . H e n c e t h e r e i s n o n o n z e r o e l e m e n t λ ∈ R s u c h t h a t λϕ = 0 . P r o b l e m 3 . S e c . 7 . 3 . 2 1 P r o o f . L e t I M b e a n i d e a l o f M n (R) . L e t J = {entries of matrices in I M } . I f a, b ∈ J , t h e n t h e r e a r e t w o m a t r i c e s A a , A b ∈ I M , s . t . , a = a pq o f A a a n d b = b rs o f A b w i t h A a = (a ij ) a n d A b = (b ij ) . T h e n a pq − b rs = c 11 i n E 1 p A a E q1 − E 1r A a E s1 = (c ij ) ∈ I M . S o a pq − b rs ∈ J i . e . , a − b ∈ J . ∀r ∈ R , ra pq = d pq i n E r A a = (d pq ) ∈ I M w h e r e E r = (a ij ) w i t h a ii = r f o r a l l i ∈ {1,...,n} a n d a ij = 0 f o r a l l i = j . S i m i l a r l y ar = a pq r = e pq i n A a E r = (e pq ) ∈ I M . S o ra,ar ∈ J . I n a l l , J i s a n i d e a l i n R , i . e . , e v e r y i d e a l o f M n (R) i s e q u a l t o M n (J ) f o r s o m e i d e a l J o f R . P r o b l e m 4 . ( S e c . 7 . 3 . 3 4 ) P r o o f . ( a ) I + J = {a + b | a ∈ I ,b ∈ J } . ∀a ∈ I , a = a +0 ∈ I + J , t h u s I ⊆ I + J . S i m i l a r l y J ⊆ I + J . S o I + J i s a n o n e m p t y s e t c o n t a i n i n g b o t h I a n d J .

exam midterm 2

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