Math 4441/6441 (Spring 2008) Homework Set #11 (Due 4/17) Assignment
Problem 1. Let Gbe a group with |G|= 70 and let a∈Gbe a (fixed) element.
(1) If a42 =e, determine all the possible value(s) of ord(a).
(2) If a42 =e,a20 6=eand a21 6=e, determine ord(a). Show your reasoning carefully.
Problem 2. Let Gbe a group and Hbe a (fixed) subgroup of G. Denote F={h3|h∈H}.
(1) If Gis abelian, show F≤G(that is, show Fis a subgroup of G).
(2) True or false: In general (i.e., without the assumption that Gis abelian), F≤G.
Give a proof or a counterexample. .............................True False
Problem 3. Let G= (R,+) be the group of all real numbers under addition and let H=
(R+,·) be the group of all positive real numbers under multiplication. Determine if each
of the following is a group homomorphism and, if so, determine if it is an isomorphism.
(1) f:G→Hdefined by f(x) = e|x|for all x∈G.
(2) g:G→Hdefined by g(x) = e−2xfor all x∈G.
Problem 4. Let G= (R,+) and H= (R+,·) be as in Problem 3 above. Determine if each
of the following is a group homomorphism. Show your reasoning.
(1) f:G→Gdefined by f(x) = 2xfor all x∈G.
(2) g:G→Gdefined by g(x) = x2for all x∈G.
(3) ϕ:H→Hdefined by ϕ(x) = x2for all x∈H.
(4) ψ:H→Hdefined by ψ(x) = 2xfor all x∈H.
Problem 5 (6441 problem).Let Gbe a group and a∈Gbe a (fixed) element. Consider Z,
which is a group under addition. Define f:Z→Gby f(n) = anfor all n∈Z.
(1) Show fis a group homomorphism from Zto G.
(2) If ord(a) = ∞, determine/describe Ker(f) explicitly.
(3) If ord(a) = k < ∞, determine/describe Ker(f) explicitly.
For the very same assignment with hints, click here.
Notice. All quoted results, such as Theorem x.y, are from the textbook A First Course in
Abstract Algebra with Applications (3rd Ed.) by J. J. Rotman, Pearson Prentice Hall.
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