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MAT301H1 - S: Groups and Symmetries
Homework 5
1 Problems to be submitted
• Make sure you follow all the indications as stated in the syllabus.
• Please take into account that there are no extensions for this homework, as it is the last one
before the midterm.
1. In lecture we have encountered the following concept repeatedly. We have not given it a proper name
however. We do it now.
Definition: Let G1, ..., Gn be groups. We define their external direct product as the new
group whose underlying set is G1 × · · · ×Gn and whose product is
(x1, ..., xn) · (y1, ..., yn) = (x1y1, ..., xnyn).
We denote this group simply by G1 × · · · ×Gn
(a) (3 points) Prove that G1 × · · · ×Gn is a group.
(b) (2 points) Prove that the subset {1} × {1} × ...× {1} ×Gi × {1} · · · × {1} (i.e. the elements whose
entries are all identities except in the i−th coordinate) is a normal subgroup of the direct product
that is isomorphic to Gi.
(c) (2 points) Determine if the following statement is true or false. If it is true provide a proof, if it is
false give a counterexample.
Statement: Let H be a subgroup of G1×G2. Then, there exist subgroups H1 ≤ G1 and H2 ≤ G2
such that H = H1 ×H2.
(d) (2 points) We now consider the following definition
Definition: Let G be a group and H,K two subgroups of G. We say G is the internal
direct product of H and K if the following three conditions are satisfied:
1. H and K are normal subgroups of G.
2. H and K have trivial intersection (this means, their only common element is the
identity).
3. Every element of G can be written in the form hk for some h ∈ H and k ∈ K.
Prove that G is the internal direct product of its subgroups H and K if and only if it is isomorphic
to H ×K.
1
(e) (1 point) Let m and n be relatively prime integers. Prove that
Cmn ∼= Cm × Cn.
2. In lecture we studied the Dihedral group D6 associated to the hexagon. We produced the following table
for the subgroups.
We did this to construct dihedral groups
In this problem we will produce similar tables (i.e. counting all subgroups) for other dyhedral groups.
We will also answer some questions about them.
For this problem let p and q be different odd prime numbers.
(a) (3 points) Make the Hasse Diagram of Dp and the table of subgroups as above. Specify which
subgroups are normal.
(b) (3 points) Make the Hasse Diagram of Dpq and the table of subgroups as above. Specify which
subgroups are normal.
(c) (3 points) Make the Hasse Diagram of Dp2 and the table of subgroups as above. Specify which
subgroups are normal.
(d) (1 point) Let n be an odd integer. Based on your previous work, give a prediction of how the Hasse
Diagram, the table of subgroups and the normal subgroups behave for Dm. You don’t have to prove
your prediction (but try to do it if you want!) but you have to explain why you predict this.
3. Consider R be the 2 × 2 × 1 Rubik’s cube group we have studied in class and in previous homeworks.
Answer the following questions about R. You should justify your answers.
(a) (3 points) How many normal subgroups of order 24 does R have?
(b) (3 points) Find all orders for which R has a normal subgroup of that order.
(c) (2 points) Find the composition series length (that is, the length of a composition series) for R and
the factor groups.
(d) (2 points) Is R an inner direct product of two of its subgroups?