This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: c ? (10 points) (7) Let G be a group, H a commutative group, ϕ : G → H a homomorphism onto H . Let N be a subgroup of G that contains Ker ϕ . Show that N is a normal subgroup of G . (10 points) 4 (8) Let H be a subgroup of a group G . Suppose g ∈ G and o ( g ) = n ∈ N . If g m ∈ H and if m and n are relatively prime, prove that g ∈ H . (7 points) (9) Let G be a group, N ≤ G , A ⊆ G . If N £ G , does AN = NA have to hold true? (8 points) 5 (10) Give an example of a Feld di±erent from Q , R , C , Z p ( p prime). (5 points) (11) Characterize all rings in which the identity x 2y 2 = ( x + y )( xy ) is valid. (5 points) 6 (12) Let A be an ideal of R . Is it true that Mat 2 ( R ) / Mat 2 ( A ) ∼ = Mat 2 ( R/A )? (10 points) 7 (13) Let D be a principal ideal domain, a, b relatively prime elements of D . Is D/abD isomorphic to D/aD ⊕ D/bD ? (15 points) 8...
View
Full
Document
This note was uploaded on 11/08/2011 for the course MATH 321 taught by Professor Ergenc during the Spring '11 term at Boğaziçi University.
 Spring '11
 ergenc
 Math

Click to edit the document details