Questions and Answers of Algebra Graduate Texts In Mathematics

If G is a finitely generated abelian group such that G /Gt, has rank n, and H is a subgroup of G such that H/ Ht has rank n-m, then m ≤ n and (G/ H)/(G/ H)t has rank n - m.
(a) Z satisfies the ACC but not the DCC on subgroups.(b) Every finitely generated abelian group satisfies the ACC on subgroups.
(a) Show that there is a nonabelian subgroup T of S3 X Z4 of order 12 generated by elements a,b such that |a| = 6, a3 = b2, ba = a-1b.(b) Any group of order 12 with generators a,b such that lal = 6,
Every subgroup and every quotient group of a nilpotent group is nilpotent.
Proof that An abelian group has a composition series if and only if it is finite.
Proof that The direct sum of a family of free abelian groups is a free abelian group. (A direct product of free abelian groups need not be free abelian).
Let G be a group acting on a set S containing at least two elements. Assume that G is transitive; that is, given any x,y ϵ S, there exists g ϵ G such that gx = y. Prove(a) for x ϵ S, the orbit x̄
If H is a normal subgroup of order pk of a finite group G, then H is contained in every Sylow p-subgroup of G.
Proof that Let k, m ϵ N*. If(k,m) = 1, then kZm = Zm andZm[k] = 0. If k|m, say m = kd, then kZm ≅ zd andZm[k] ≅ zk.
Let F be the free group on the set X, and let Y ⊂ X. If His the smallest normal subgroup of F containing Y, then F/ His a free group.
If ∫: G → His a homomorphism, a ϵ G, and∫(a) has finite order in H, then lal is infinite or I∫(a)I divides lal.
A pointed set is a pair (S,x) with S a set and x ϵ S. A morphism of pointed sets (S,x) → (S',x') is a triple (∫,x,x'), where ∫: S →S' is a function such that ∫(x) = x'. Show that pointed
S3 is not the direct product of any family of its proper subgroups. The same is true of Zpn (p prime, n ≥ 1) and z.
If ∫: G → His a homomorphism of groups, then∫(eG) = e11 and∫(a-1) = ∫(a)-1 for all a ϵ G. Show by example that the first conclusion may be false if G, H are monoids that are not groups.
Find four different subgroups of S4 that are isomorphic to S3 and nine isomorphic to S2.
Every nonidentity element in a free group F has infinite order.
Let G be a group and {Hi I i ϵ} a family of subgroups. Then for any a€ G, (H)a= H,a. n
Let a,b be elements of group G. Show that lal = la-1l; labl = lbal, and lal = lcac-1l for all c ϵ G.
Proof that If N is a subgroup of index 2 in a group G, then N is normal in G.
If {Ni, Ii ϵ /} is a family of normal subgroups of a group G, then. is a ial normal subgroup of G. iel
Let Io = Ø and for each n e N* let In = { 1,2,3, ... , n}.(a) In is not equipollent to any of its proper subsets(b) Im and In are equipollent if and only if m = n.(c) Im is equipollent to a subset
If ∫ : A → B is an equivalence in a category e and g : B → A is the morphism such that g о ∫= 1A,∫ о g = 1b, show that g is unique.
(a) Let H be the cyclic subgroup (of order 2) of S3 generated byThen no left coset of H (except H itself) is also a right coset. There exists a ϵ S3 such that aH ∩ Ha= {a}.(b) If K is the cyclic
Give an example of groups Hi, Ki such that H1 X H2 ≅ K1 X K2 and no Hi is isomorphic to any Ki.
A group G is abelian if and only if the map G → G given by x |→ x-1 is an automorphism.
Let G be an (additive) abelian group with subgroups H and K. Show that G ≅ H ⊕ K if and only if there are homomorphisms such that π1L1 = 1H, π2L2 = 1K, π1L2 = 0 and
(a) Prove that Sn is generated by then - 1 transpositions (12), (13), (14), ... , (In).(b) Prove that Sn is generated by the n - 1 transpositions (12), (23), (34), ... , (n - 1n).
Show that the free group on the set {a} is an infinite cyclic group, and hence isomorphic to Z.
Let Q8 be the group ( under ordinary matrix multiplication) generated by the complex matriceswhere i2 = -1. Show that Q8 is a nonabelian group of order 8. Q8 is called the quaternion group. 0
Let G be an abelian group containing elements a and b of orders m and n respectively. Show that G contains an element whose order is the least common multiple of m and n.
A lattice (A, ≤) (see Exercise 1) is said to be complete if every nonempty subset of A has both a least upper bound and a greatest lower bound. A map of partially ordered sets f: A → B is said to
(a) Every infinite set is equipollent to one of its proper subsets.(b) A set is finite if and only if it is not equipollent to one of its proper subsets [see Exercise I].
Let G be a group (written additively), Sa nonempty set, and M(S,G) the set of all functions ∫: S → G. Define addition in M(S,G) as follows: (∫ + g) : S → G is given by sI→ ∫(s) + g(s) ϵ
The following conditions on a finite group Gare equivalent.(i) I GI is prime.(ii) G ≅ (e) and G has no proper subgroups.(iii) G ≅ Zp for some prime p.
Let H be the group (under matrix multiplication) of real matrices generated byShow that His a nonabelian group of order 8 which is not isomorphic to the quaternion group of Exercise 3, but is
Let F be a free group and let N be the subgroup generated by the set {xn| x ϵ F, n a fixed integer}. Show that N ⊲ F.
Let G be an abelian group of order pq, with (p,q) = 1. Assume there exist a,b ϵ G such that lal = p, lbl = q and show that G is cyclic.
Let N be a subgroup of a group G. N is normal in G if and only if (right) congruence modulo N is a congruence relation on G.
Exhibit a well ordering of the set Q of rational numbers.
Prove that,(a) Z is a denumerable set.(b) The set Q of rational numbers is denumerable.
Proof that Let α be an integer and pa prime such that P X α. Then aP-1 = 1 (mod p).
In the category α of abelian groups, show that the group· A1 X A2, together with the homomorphisms L1 : A1 → A1 X A2 and L2 : A2 → A1 X A2 (as in the Example preceding
Give an example to show that the weak direct product is not a coproduct in the category of all groups.
Let ∼ be an equivalence relation on a group G and let N = {a ϵ G I a ∼ e}. Then ∼ is a congruence relation on G if and only if N is a normal subgroup of G and ∼ is congruence modulo N.
Let S be a set. A choice function for S is a function f from the set of all nonempty subsets of S to S such that ∫(A) ϵ A for all A ≠ Ø, A ⊂ S. Show that the Axiom of Choice is equivalent to
Proof that If A,A',B,B' are sets such that lAl = IA'I and lBl = IB'I, then IA X Bl = IA' X B'I. u in addition A ∩ B = Ø = A' ∩ B', then IA ∪ Bl = IA' ∪ B'I- Therefore multiplication
Write out a multiplication table for the group D4 *
Prove that there are only two distinct groups of order 4 (up to isomorphism), namely Z4 and Z2⊕Z2.
Proof that Every family { Ai |i ϵ I} in the category of sets has a coproduct. U Ai, = { (a,i) ε ( U Ai)} X I |a ε Ai} with A; → U Ai given by a ↦ (a,i). U Ai, is called the disjoint union of
Proof that Let G, H be finite cyclic groups. Then G X H is cyclic if and only if(IGl,IHI) = 1.
Let S be a nonempty subset of a group G and define a relation on G by a ∼ b if and only if ab-1 ϵ S. Show that ∼ is an equivalence relation if and only if S is a subgroup of G.
Proof that The group defined by generators a, b and relations a8 = b2a4 = ab-1ab = e has order at most 16.
Let N < S4 consist of all those permutations σ such that σ(4) = 4. Is N normal inS4?
Let S be the set of all points (x,y) in the plane with y ≤ 0. Define an ordering by (x1,Y1) ≤ (x2,Y2) = x1 = x2 and Y1 < Y2, Show that this is a partial ordering of S, and that S has
For all cardinal numbers α, β̃, γ(a) α + β̃ = β̃ +α and αβ̃ = β̃α (commutative laws).(b) (α + β̃) + γ = β̃ + (α + γ) and (αβ̃ = α(β̃γ) (associative
Prove that the symmetric group on n letters, Sn, has order n!.
Let H,K be subgroups of a group G. Then HK is a subgroup of G if and only if HK= KH.
Every finitely generated abelian group G ≠ (e) in which every element (except e) has order p (p prime) is isomorphic to Zp ⊕Zp·· •⊕ZP (n summands) for some n ≥ 1.
Prove that if all the sets in the family {AiIi ε I ≠ Ø} are nonempty, then each of the projectionsis surjective. πk: ΠΑ II A₁ → Ak
A nonempty finite subset of a group is a subgroup if and only if it is closed under the product in G.
An is the only subgroup of Sn of index 2.
The cyclic group of order 6 is the group defined by generators a,b and relations a2 = b3 = a-1b-1ab = e.
Proof that If G is a cyclic group of order n and k I n, then G has exactly one subgroup of order k.
Let H < G; then the set aHa-1 is a subgroup for each a ϵ G, and H ≅ aHa-1.
Let In be as in Exercise 1. If A ∼ Im and B ∼ In and A ∩ B = Ø, then (A U B) ∼ Im+n and A X B "'Imn. Thus if we identify IAI with m and IBI with n, then IAI + IBI = m + n and IAIIBI = mn.
Write out an addition table for Z2⊕Z2, Z2⊕Z2 is called the Klein four group.
Let G be a group of order pkm, with p prime and (p,m) = 1. Let H be a subgroup of order pk and K a subgroup of order pd, with O < d ≤ k and K ⊄ H. Show that HK is not a subgroup of G.
Let F be a free object on a set X (i: X → F) in a concrete category e. If e contains an object whose underlying set has at least two elements in it, then i is an injective map of sets.
Let H,K,N be nontrivial normal subgroups of a group G and suppose G = H X K. Prove that N is in the center of G or N intersects one of H,K nontrivially. Give examples to show that both possibilities
Let p be prime and Ha subgroup of Z(p∞).(a) Every element of Z(p∞ ) has finite order pn for some n ≥ 0.(b) If at least one element of H has order pk and no element of H has order greater than
Show that N = {(1), (12)(34), (13)(24), (14)(23)} is a normal subgroup of S4 contained in A4 such that S4/ N ≅ S3 and A4/ N ≅ Z3.
Show that the group defined by generators a,b and relations a2 = e, b3 = e is infinite and nonabelian.
Proof that Let G be a finite group and-Ha subgroup of G of order n. If H is the only subgroup of G of order n, then His normal in G.
Let (A, ≤) be a linearly ordered set. The immediate successor of a ε A (if it exists) is the least element in the set {x ε A I a < x} Prove that if A is well ordered by ≤ then at most one
If A ∼ A', B ∼ B' and ∫: A →B is injective, then there is an injective map A' → B'. Therefore the relation ≤ on cardinal numbers is well defined.
Proof that:(a) The relation given by a ∼ b ⇔ a - b ϵ Z is a congruence relation on the additive group Q.(b) The set Q/Z of equivalence classes is an infinite abelian group.
If H and K are subgroups of finite index of a group G such that [G: H] and [G: K] are relatively prime, then G = HK.
Suppose X is a set and F is a free object on X (with i : X → F) in the category of groups the existence of F is proved. Prove that i(X) is a set of generators for the group F.
Corollary 8. 7 is false if one of the Ni is not normal. Data from in Corollary 8. 7
The set {σ ϵ Sn I σ(n) = n} is a subgroup of Sn which is isomorphic to Sn-I·
Show that the group A4 has no subgroup of order 6.
The group defined by generators a, b and relations an = e (3 ≤ n ϵ N*), b2 = e and abab = e is the dihedral group Dn. [See Theorem 6.13.]Theorem 6.13For each n ≥ 3 the dihedral group Dn is a
Let {Ni| i ϵ I| be a family of subgroups of a group G. Then G is the internal weak direct product of {Ni| i ϵ I} if and only if: (i) ai,ai = aiai for all i ≠ j and ai ϵ Ni, ai ϵ Ni; (ii) every
For n ≥ 3 let Gn be the multiplicative group of complex matrices generated bywhere i2 = -1. Show that Gn ≅ Dn. 0 1) and y e²mi/n 0 0 e-2mi/n
Show that a group that has only a finite number of subgroups must be finite.
All subgroups of the quaternion group are normal (Exercises 2.3 and 4.14). Exercises 2.3Let N be a subgroup of a group G. N is normal in G if and only if (right) congruence modulo N is a
How to prove that every infinite subset of a denumerable set is denumerable.
Let p be a fixed prime. Let RP be the set of all those rational numbers whose denominator is relatively prime top. Let Rp be the set of rationals whose denominator is a power of p (pi, i ≥ 0).
Proof that If H, K and N are subgroups of a group G such that H < N, then prove that HK ∩ N = H(K ∩ N).
Proof that If a group G is the (internal) direct product of its subgroups H, K, then H ≅ G/ K and G/H ≅ K.
Let ∫: G → H be a homomorphism of groups, A a subgroup of G, and B a subgroup of H.(a) Ker ∫and ∫-1(B) are subgroups of G.(b) ∫(A) is a subgroup of H.
Let p be a prime and let Z(p∞) be the following subset of the group Q/ZShow that Z(p∞) is an infinite group under the addition operation of Q/Z. Z(p") = (a/b & Q/Z a,be Z and b = pi for some i
If {Gi | i ϵ I} is a family of groups, then is the internal weak direct product its subgroups {Li(Gi)| i ϵ I}. II"G.
The group defined by the generator b and the relation bm = e (m ϵ N*) is the cyclic group Zm.
(a) If G is a group, then the center of G is a normal subgroup of G (see Exercise 2.11);(b) the center of Sn is the identity subgroup for all n > 2.Exercise 2.11If His a cyclic subgroup of a group
Proof that If G is an abelian group, then the set T of all elements of G with finite order is a subgroup of G.
The infinite set of real numbers R is not denumerable (that is, NO < IRI). it suffices to show that the open interval (0,1) You may assume each real number can be written as an infinite decimal.
Let H, K, N be subgroups of a group G such that H < K, H ∩ N = K ∩ N, and HN = KN. Show that H = K.
List all subgroups of Z2⊕Z2. Is Z2⊕Z2 isomorphic to Z4?
Let a be the generator of order n of Dn. Show that (a) ◁ Dn and Dn/(a) ≅ Z2.

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