Math 4441/6441 (Spring 2008) Homework Set #11 (Due 4/17) Hints
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.
Hint. (1). What can be said about ord(a) from |G|= 70 and a42 =e?
(2). What can be said about ord(a) from |G|= 70, a42 =e,a20 6=eand a21 6=e?
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
Hint. An element x∈Gis in Fif and only if xcan be written as x=h3for some h∈H.
(1). Why is ein F? (Refer to the last paragraph/line.) Given x, y ∈F, how do you show
xy ∈Fand x−1∈F? (Refer to the last paragraph.)
(2). If you believe this is false, then give an explicit counterexample. Of course, such
counterexample, if it exists, must be non-abelian because of part (1). What is ‘the most
frequently used non-abelian group’ in our course? Study it and all its subgroups.
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.
Hint. Apply the definition of homomorphisms. Be aware of the operations in Gand H.
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.
Hint. Apply the definition of homomorphisms. Be aware of the operations in Gand 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.
Hint. Recall that Ker(f) = {n∈Z|f(n) = e}. So, for (2) and (3), you could start by listing
‘all’ integers nsuch that f(n) = e. Remember that, when ord(a)<∞,an=eif and only if
ord(a) divides n.
For solutions, 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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