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Exam I Review Sheet Math 4201

The first exam will be on Friday, February 17, 2006. The syllabus for Exam I is Chapter 7 and Section 8.1 in the text, together with the homomorphism and cyclic handouts. Some of the main examples and facts from this material are listed below. All of Chapter 7 is listed, since the material is cumulative and the concepts and results in the sections on which we concentrated are dependent on examples, definitions, and results from earlier in the Chapter and from earlier Chapters that you will have covered last semester. Following are some of the concepts and results you should know:

∗ • Examples of groups such as Sn, Dn, GL(n, R), Z, Zn, Zn = Un = group of units of the integers modulo n, cyclic groups. In particular, know what elements of these groups are and what the group operation is in each case. Know the order of each of the groups which is finite.

• What does it mean to be a ? Know the criterion to be a subgroup and how to use it to check that H is a subgroup of a group G: If G is a group, then a nonempty H ⊆ G is a subgroup if and only if a, b ∈ H implies ab ∈ H and a ∈ H implies a−1 ∈ H.

• What is an abelian group?

• What is a ? What is a generator of a cyclic group?

• What is the condition for an element [a]n ∈ Zn to have a multiplicative inverse? (Answer: a and n should be relatively prime integers.

• If G = hgi is a cyclic group of order n, know the condition for gk to be a generator of G. Answer: gk is a generator of G if and only if k and n are relatively prime. See Theorem 7, Page 3 of the Cyclic Group Supplement.

• What is the order of a group (denoted |G|)? What is the order o(g) of an element g ∈ G? If o(g) = m, then an = e if and only if m | n.

• Every subgroup of a cyclic group is cyclic (Theorem 9 of Cyclic Group Supplement).

• Know the Fundamental Theorem of Finite Cyclic groups (Theorem 10 of the Cyclic Group Supplement.)

• Know how to use the Fundamental Theorem of Finite Cyclic groups to determine all the of a cyclic group and draw the diagram.

• The right (left) cosets of a subgroup H are the sets Ha = {ha : h ∈ H} (aH = {ah : h ∈ H}). −1 They are precisely the equivalence classes of the equivalence a ≡H b ⇐⇒ ab ∈ H −1 (a ≈H b ⇐⇒ a b ∈ H). The set of right cosets of H in G is denoted G/H. • Lagrange’s Theorem: |G/H| = |G|/|H|. That is, the number of cosets of H is exactly |G|/|H|. In particular, |H| | |G|.

• The order of every element of a group divides the order of the group.

• If the order of G is prime, then G is cyclic.

• Know the meaning of the term isomorphism.

• Sn is the group of permutations of the set {1, 2, . . . , n}. |Sn| = n!

1 Exam I Review Sheet Math 4201

• An is the group of even permutations in Sn. |An| = (n!)/2 • Every permutation is a product of disjoint cycles.

• A transposition is a cycle of length 2. Every permutation is a product of transpositions. The number of transpositions in such a product for a permutation σ is always even or always odd. σ is even if it is a product of an even number of transpositions; σ is odd if it is a product of an odd number of transpositions.

• An r-cycle (j1, j2, . . . , jr) is an even permutation if r is odd and it is odd if r is even. This follows from the factorization

(j1, j2, . . . , jr) = (j1, jr)(j1, jr−1) ··· (j1, j2).

• A group G is cyclic if G = hai, i.e., if every element of G is a power of a.

• If a ∈ G, then the order of a, denoted |a|, is the smallest natural number n such that an = e.

• Lagrange’s Theorem: If H is a subgroup of G, then |H| divides |G|, and |G|/|H| = [G : H] is the number of left cosets of H in G. It is also the number of right cosets.

• If a ∈ G, then |a| divides |G|.

• If a ∈ G, then am = e if and only if |a| divides m.

• If σ = γ1γ2 ··· γk is the disjoint cycle factorization of σ ∈ Sn, then |σ| = lcm(|γ1|,..., |γk|). • If |G| is prime, then G is cyclic.

• The subgroups of Z are the nZ = {nm : m ∈ Z}.

• A function f : G → H between groups G and H is a homomorphism if f(ab) = f(a)f(b). Some properties of homomorphisms are: f(1) = 1, f(a−1) = (f(a))−1, f(ak) = (f(a))k for all k ∈ Z.

• Let H be a subgroup of a group G and let a, b ∈ G. Then the following are properties of the cosets of H:

1. Ha = H if and only if a ∈ H. 2. Ha = Hb if and only if ab−1 ∈ H. 3. If a ∈ Hb, then Ha = Hb. 4. Either Ha = Hb or Ha ∩ Hb = ∅. 5. The distinct right (left) cosets of H are a partition of G, and if the order of G is finite, then the number of right (left) cosets is |G|/|H|.

• If G is a group and H is a subgroup, then H is a if gH = Hg for all g ∈ G.

• A subgroup H of G is normal if and only if gHg−1 ⊆ H for all g ∈ G.

• If G is a group, then then subgroups {1}, G and Z(G) = the of G, are always normal subgroups of G.

• If G is abelian, then every subgroup of G is normal in G.

2 Exam I Review Sheet Math 4201

• If H is a subgroup of index 2 in G, then H is normal in G.

• A group G is simple if G 6= {1} and the only normal subgroups of G are {1} and G.

• An abelian group G 6= {1} is simple if and only if it is cyclic of prime order.

• The alternating group An is a simple group if n ≥ 5. • If K is a normal subgroup of G, then Ka · Kb = Kab is a well defined multiplication of right (=left) cosets. In this case the set G/K = {Ka : a ∈ G} of right (=left) cosets of K forms a group under this multiplication. Some of the properties of this group G/K are:

1. The group operation is (Ka)(Kb) = Kab. 2. The identity of G/K is the coset K. 3. The inverse of Ka is Ka−1, i.e., (Ka)−1 = Ka−1. 4. The exponent rule in G/K is (Ka)n = Kan. 5. The order of Ka is the smallest positive power n such that an ∈ K. 6. The mapping ϕ : G → G/K defined by ϕ(a) = Ka is a surjective group homomorphism. 7. If G is abelian, then G/K is abelian. 8. If G = hai is cyclic, then G/K is also cyclic; in fact G/K = hKai 9. If G is finite, then |G/K| = |G|/|K| = [G : K].

• The direct product G1 × G2 × · · · × Gn of groups G1, G2, ..., Gn is the set of n-tuples (g1, g2, . . . , gn) where gi ∈ Gi with the group operation defined componentwise:

(g1, g2, . . . , gn)(h1, h2, . . . , hn) = (g1h1, g2h2, . . . , gnhn).

• If G1, G2, ..., Gn are groups, their direct product G is a group of order |G1| |G2| · · · |Gn|. This means that if any Gi is infinite, then so is G.

• If G = G1 × G2 × · · · Gn and g = (g1, g2, . . . , gn), then the order of g is

o(g) = lcm {o(g1), o(g2), . . . , o(gn)} .

• Know how to recognize if a group G is the (internal) direct product of subgroups H and K. This is the content of the following recognition theorem from Section 8.2: Suppose that G is a group with subgroups H and K such that

1. H and K are normal in G, 2. H ∩ K = hei, and 3. G = HK = {hk : h ∈ H, k ∈ K} .

Then G =∼ H × K.

Review Exercises

Be sure that you know how to do all assigned homework exercises. The following are a few supple- mental exercises similar to those already assigned as homework. These exercises are listed randomly. That is, there is no attempt to give the exercises in the order of presentation of material in the text, and there is no claim that a representative of every assigned exercise is included.

3 Exam I Review Sheet Math 4201

1. If G is a group and a, b ∈ G, prove the following facts. (a) |a−1| = |a| (b) |b−1ab| = |a| (c) |ba| = |ab|

∗ 2. Show that if G = Z15 = U15 is the group of units of Z15, then G = h2, 13i. Is G cyclic? ½· ¸¯ ¾ 1 n ¯ 3. Prove that H = ¯ n ∈ Z is a cyclic subgroup of GL(2, R). 0 1 ¯

4. Explain why Z6 and S3 are not isomorphic. 5. Let G = hai where |a| = 30 and let H = ha4i. What is |G/H|? 6. Let G = hai and let H = ha12, a20i. Find a generator for H and compute |H|.

7. Let G = Z4 × Z4 and let H be the cyclic subgroup generated by (3, 2). Show that G/H is isomorphic to Z4. 8. If H is a normal subgroup of G and a2 ∈ H for all a ∈ G, prove that every nonidentity element of G/H has order 2. Does this mean that |G/H| = 2? ¡ ¢ ¡ ¢ 9. Verify that σ = 1 2 3 4 and τ = 2 4 generate a subgroup of S4 of order 8. 10. Show that if σ is a 6 cycle, then σ2 is the product of two disjoint 3 cycles. If τ is a 5 cycle, what is the disjoint cycle factorization of τ 2? 11. Let G = hai × hbi where |a| = 8, |b| = 2, and K = h(a2, b)i. Write all the cosets in G/K and describe the group G/K. 12. (a) If a, b, and c are elements of a group G, then solve the equation axb = c for x ∈ G. µ ¶ µ ¶ 1 2 3 4 5 1 2 3 4 5 (b) Let α = , β = . Solve each of the following equations 2 5 1 3 4 5 1 3 2 4 in S(5). i. Solve for γ: αγ = β. ii. Solve for δ: δα = β. iii. Solve for ν: ανα−1 = β. 13. Let G = hgi be a cyclic group of order 225 = 9 × 25. List all of the subgroups of G (by giving an appropriate generator) and draw the lattice of subgroups diagram. 14. Suppose that G is the group defined by the following multiplication table. · a b c d e f g h a a b c d e f g h b b a h g f e d c c c d e f g h a b d d c b a h g f e e e f g h a b c d f f e d c b a h g g g h a b c d e f h h g f e d c b a

4 Exam I Review Sheet Math 4201

(a) Which element is the identity of G? (b) What is g−1? (c) Find C(g) = {x ∈ G : xg = gx}. (d) Find the order of d and the order of g.

15. (a) If G is a group with |G| = 21, what are the possible orders of elements of G? (b) If |G| = 21, and H is a subgroup of G other than G itself, explain why H must be cyclic. (c) Let K be a subgroup of a group L and let L be a subgroup of a group M. What are the possible values of |L| if |K| = 6 and |M| = 72?

16. Determine all the left cosets of H in G if:

(a) G = Z24, H = h[4]24i. ¡ ¢ (b) G = S3, H = h 2 3 i. 17. Let H = 5Z in Z. Then H is a subgroup of the additive group Z. Determine whether the following cosets of H are the same. (Remember the group operation is + so cosets will be written a + H, rather than aH.)

(a) 12 + H and 27 + H (b) 13 + H and −2 + H (c) 126 + H and −1 + H

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