## Transcribed Text

Let n € S8 be the permutation defined by
(1) = 3,
(2) = 8,
(3) = 5,
(4) = 4,
TT(5) = 6,
(6) = 1,
\T(7) = 7,
(8) = 2.
a. Prove that T has order 4.
b. Let G be the subgroup fo S8 generated by TT, i.e., G = (IT) = {e, 11, T2,,T3}.
Describe all of the orbits of G, as was done in Example 6.11.
c. Let X = {1, 2, 3, 4, 5, 6, 7, ,8}, so G acts on X. For each k € X, describe the stabilizer Gk.
d. Use your data from (b) and (c) to explicitly verify the orbit-stabiliser counting formula
Exercises
Section 6.1. Normal Subgroups and Quotient Groups
6.1. Let : G ! G0 be a homomorp hhiissmm
(a) Prove that the image (G) = (g) : g 2 G }is a subgroup of G0.
(b) Suppose that G is a finite group. Prove that
#G = # (G0) #ker( ):
6.2. Let G be a group, let H G and K G be subgroups, and assume that K is a normal
subgroup of G.
(a) Prove that HK = fhk : h 2 H; k 2 Kg is a subgroup of G.
(b) Prove that H \K is a normal subgroup of H, and that K is a normal subgroup of HK.
(c) Prove that HK=K is isomorphic to H=(H \ K). (Hint. What is the kernel of the surjective
homoorphism H ! HK=K?)
6.3. Let G be a group, let K H G be subgroups, and assume that K is a normal
subgroup of G.
(a) Prove that H=K is naturally a subgroup of G=K.
(b) Conversely, prove that every subgroup of G=K looks like H=K for some subgroup H
satisfying K H G.
Section 6.2. Groups Acting On Sets and a Counting Theorem
6.4. Let G be a group, let X be a set on which G acts, and let x 2 X. Prove that the
stabilizer Gx of x is a subgroup of G.
6.5. Let p be a prime. We proved in Corollary 6.18 that a group with p2 elements must be
abelian. Let G be a group with p3 elements.
(a) Mimic the proof of Corollary 6.18 to try to prove that G is abelian. Where does the proof
go wrong?
(b) Give two examples of non-abelian groups with 23 elements. (This shows that the proof
in (a) can’t work.)
6.6. Let G be a group, let X be a set, and let SX be the symmetry group of X as defined in
Example 2.14. Let
: G ! SX
be a function from G to SX, and for g 2 G and x 2 X, let g x = (g)(x). Prove that this
defines a group action if and only if the function is a group homomorphism.
Section 6.4. Sylow’s Theorem
6.7. Let p be prime, let n 1, let a 1, and consider the number
A = a(a + 1)(a + 2) (a + pn 1):
Factor A as A = pkB with p - B.
(a) Prove that k does not depend on the value of a; cf. Lemma 6.20.
(b) Find a simple closed formula for k in terms of p and n.
6.8. This exercise asks you to give two different proofs of the following stronger version of
the first part of Sylow’s Theorem. Theorem: Let G be a finite group, let p be a prime, and
suppose that #G is divisible by pr. Prove that G has a subgroup of order pr. (Note that pr is
not required to be the largest power of p that divides G.)
(b) Give a proof by induction on the power of p that divides#G. Use the fact that we already
proved that G has a p-Sylow subgroup H. Then use Theorem 6.17 to deduce that H has
a non-trivial center Z(H), and apply the induction hypothesis to H=Z(H).
6.9. This exercise descrbes a way to create new groups from known groups. Let G be a group.
An isomorphism from G to itself is called an automorphism of G. The set of automorphisms
is denoted
Aut(G) = fgroup isomorphisms G ! Gg:
We define a compostion law on Aut(G) as follows: for ; 2 Aut(G), we define to be
the map from G to G given by ()(g) =
(g)
.
(a) Prove that this composition law makes Aut(G) into a group.
(b) Let a 2 G. Define a map a from G to G by the formula
a : G ! G; a(g) = aga1:
Prove that a 2 Aut(G), and that the map
G ! Aut(G); a 7! a; (6.24)
is a group homomorphism.
(c) Prove that the kernel of homomorphism (6.24) is the center Z(G) of G.
(d) Elements of Aut(G) that are equal to a for some a 2 G are called inner automorphisms,
and all other elements of Aut(G) are called outer automorphisms. Prove that G
is abelian if and only if its only inner automorphism is the identity map.
6.10. Let Cn be a cyclic subgroup of order n, and let Aut(Cn) be the automorphism group
of Cn; see Exercise 6.9. Prove that Aut(Cn) is isomorphic to (Z=nZ), the group of units in
the ring Z=nZ. Hint. Show that the following map is a group isomorphism:
(Z=nZ) ! Aut(Cn); k mod n 7! k; where k(g) = gk.
6.11. In Example 6.24 we showed that there are exactly two groups of order 10. Do a similar
calcuation to find all groups of order 15.
6.12. Let G be a finite group of order #G = pq, where p and q are primes satisfying p > q.
Assume further that p 6 1 (mod q).
(a) Prove that G is an abelian group. (Hint. Example 6.25 provides a starting point for the
proof.)
(b) Prove that G is a cyclic group.

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