MAT301 Assignment 2
Show work in all problems.
(1) Let E(n) be the group of all symmetries of Rn ( not necessarily
fixing 0).
Prove that E(n) = {f : Rn → Rn | f(u) = Au + b where A ∈
O(n), b ∈ Rn}.
(2) Let G be a group and let a, b, c ∈ G such that a5 = b4 = c7 = e.
Express ((a2b−3)−2c3)−2 as a product of positive powers of a, b, c.
(3) Let G be a group such that every non identity element of G has
order 2.
Prove that G is abelian.
(4) Suppose a subset H in a group G has the property that for any
a, b ∈ H implies that b−1a−1 ∈ H also.
Does this imply that H is a subgroup of G?
Hint: Look at G = Zm for small m.
(5) Let (G, ·) be a group and. H,K ⊂ G be subgroups.
Recall that HK = {h · k | h ∈ H, k ∈ K}.
Is HK always a subgroup of G? If yes, prove it. If not, give a
counterexample.
(6) (5 pts) Let G = (Z,+) and let H = 〈18, 24, 42〉 be the subgroup
generated by 18, 24, 42.
Find d > 0 such that H = 〈d〉.
(7) Let l be a line through 0 in R2 and α > 0 be any angle. Let Fl be
the reflection nin l and Rα be the rotation by α around 0.
Let H = 〈Rα, Fl〉 ⊂ O(2) be the subgroup of O(2) generated by
Rα and Fl.
Prove that H = {F kl Rnα | k = 0, 1, n ∈ Z}.
Hint: Show that Fl ◦Rα = R−α ◦ Fl.
(8) Let G = SL(2,R) and let A =
(
0 1
−1 2
)
.
Find the centralizer of A in G.
(9) Let n ≥ 2 and let H ⊂ Dn be a subgroup. Prove that either every
element of H is a rotation or exactly half of the elements of H are
rotations.
1
2Hint: Use that the product of two rotations is a rotations, the
product of two reflections is a rotation and the product of a rotation
and a reflection is another reflection.

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