math 435 homework4

# math 435 homework4 - Homework 5 for MATH 435 Solutions...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 5 for MATH 435 Solutions Problem 1 (a) Show that ( Q , +) is not cyclic. Solution. For any q = a/b ∈ Q , a , b ∈ Z relatively prime, < q > = { ma/b : m ∈ Z } . Then 1 / ( b + 1 ) < < q > , hence Q is not cyclic. (b) Does there exist finite X ⊆ Q such that < X > = Q (as an additive group)? Solution. Let q 1 = a 1 /b 1 , . . . , q n = a n /b n ∈ Q , a i , b i ∈ Z relatively prime. Then < q 1 , . . . , q n > ⊆ { m/ ( b 1 ··· b n ): m ∈ Z } , and hence 1 / ( b 1 ··· b n + 1 ) < < q 1 , . . . , q n > . Problem 2 Let G be a group and H a subgroup. Let x ∈ G . Let xHx- 1 be the subset of G consisting of all elements xyx- 1 , y ∈ H . Show that xHx- 1 is a subgroup of G Solution. xHx- 1 inherits associativity from G . Since H is a subgroup, 1 ∈ H , and hence 1 = x 1 x- 1 ∈ H . Finally, given y ∈ xHx- 1 , again since H is a subgroup, y- 1 ∈ H , and hence xy- 1 x- 1 ∈ H . xy- 1 x- 1 is the inverse of xyx- 1 , because ( xyx- 1 )( xy- 1 x- 1 ) = xyx- 1 xy...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

math 435 homework4 - Homework 5 for MATH 435 Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online