# MATH1022 TUTORIAL EXERCISES II 1. Find all subgroups of D4_ the

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```					                           MATH1022
TUTORIAL EXERCISES II

1. Find all subgroups of D4 , the group of symmetries of a square (see
Exercises I Qu 2).
2. Find all subgroups of the group G = {1, 3, 5, 9, 11, 13} under × mod 14
(see Tutorial Exercises I Qu.1).
3. Show that the set of all complex numbers of modulus 1 forms a subgroup
of the group of non-zero complex numbers under multiplication.
4. Prove that the intersection of two subgroups of a group G is also a
subgroup of G. Let nZ be the set of all integers which are divisible by the
positive integer n. Prove that nZ is a subgroup of Z, and show that the
intersection of mZ and nZ is equal to lZ where l is the least common multiple
of m and n.
5. Describe all rotations of a cube. (Hint; there are ﬁve diﬀerent kinds of
rotations: the identity, rotations through 2π/3 and π/2, and two diﬀerent kinds
of rotation through π.) Calculate the order of the group CU B of rotations of
a cube, and ﬁnd the orders of all its elements.
6. Prove that a group G is abelian if and only if for every a, b ∈ G and
positive integer n, (ab)n = an bn .
a b
7. Show that the set of all non-singular upper triangular matrices
0 c
with real a, b, and c, forms a subgroup of GL2 (R). Generalize to GLn (R).

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