Midterm review problems
These are a few problems to help you prepare for the midterm. You don’t have to turn them in.
1. Suppose that H is a subgroup of G, that N is a normal subgroup of G, that H ∩N = {e}, and HN = G.
(a) For all h ∈ H, let ρh : N → N be the map ρh (n) = h−1 nh. Show that ρh ∈ Aut(N ) for all h ∈ H
and that the map ρ : H → Aut(N ) given by ρ(h) = ρh is a homomorphism.
(b) Show that G ∼
= H nρ N .
2. Semi-direct products can be used to construct many examples of groups:
(a) Show that Aut(Z11 ) ∼
= Z10 .
(b) Use semi-direct products to construct a non-abelian group with order 55.
3. Let Q = {1, −1, i, −i, j, −j, k, −k} be the group of order 8 with multiplication table:
×
1
−1
i
−i
j
−j
k
−k
1
1
−1
i
−i
j
−j
k
−k
−1
−1
1
−i
i
−j
j
−k
k
−i
−i
i
1
−1
k
−k
−j
j
i
i
−i
−1
1
−k
k
j
−j
j
j
−j
k
−k
−1
1
−i
i
−j
−j
j
−k
k
1
−1
i
−i
k
k
−k
−j
j
i
−i
−1
1
−k
−k
k
j
−j
−i
i
1
−1
So i2 = j 2 = k 2 = ijk = −1. We use this notation for the elements so that “−” acts like it usually
does. That is, for any a, b ∈ Q, we have −a = (−1)a and
ab = (−a)(−b) = −(−a)(b) = −(a)(−b).
(a) Show that {−1, 1} is a normal subgroup of Q. What are the elements of Q/{−1, 1}? Show that
Q/{−1, 1} is isomorphic to Z2 × Z2 .
(b) List all the subgroups of Q. Are any of them isomorphic to Q/{−1, 1}? (This problem shows that
not every quotient of a group is isomorphic to a subgroup of that group.)
4. Let G be an abelian group. Show that any quotient group of G is abelian.
5. Suppose that f : G → H is a homomorphism and a ∈ G. If o(a) = 15, what are the possible values of
o(f (a))?
6. Let G be a group and let A, B ⊂ G be normal subgroups such that A ∩ B = {e}. Show that G is
isomorphic to a subgroup of G/A × G/B.
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