MATH 3230: Abstract Algebra Spring 2024 HOMEWORK 4
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MATH 3230: Abstract Algebra
Spring 2024
HOMEWORK 4
Due at the beginning of class, Thursday, March 7
(1) For each of the following groups G and subgroups H, ind all distinct left cosets of H in G. You do not need to write out aH for every a 2 G (there will be repeats); you can stop once you know you’ve found all distinct cosets.
(a) G = Z12 , H = f0, 3, 6, 9g
(b) G = U(15) = f1, 2, 4, 7, 8, 11, 13, 14g, H = f1, 4g
(c) G = S4 , H = f(), (23), (24), (34), (234), (243)g
(2) Recall the converse of Lagrange’s Theorem, which is not true in general: If G is a group and k divides jGj, then G has a subgroup of order k. Prove that this statement is true if G is cyclic.
(3) Let G1 and G2 be isomorphic groups.
(a) Prove that if G1 is Abelian, then G2 must be Abelian.
(b) Prove or disprove: The groups S3 and U(9) are isomorphic. If you need to show they are isomorphic, you must deine a function from one to the other and show it’s an isomorphism.
(4) Let G1 and G2 be isomorphic groups.
(a) Prove that if G1 is cyclic, then G2 must be cyclic.
(b) Prove or disprove: The groups U(7) and U(9) are isomorphic. If you need to show they are isomorphic, you must deine a function from one to the other and show it’s an isomorphism.
(5) Let G1 and G2 be isomorphic groups. Prove that if G1 has a subgroup of order n, then G2 has a subgroup of order n.
2024-03-03