Ideals and Factor Rings Ring Homomorphisms Polynomial Rings Factorization of Polynomials Divisibility in Integral Domains Fields Fields Extension Fields Algebraic Extensions Finite Fields Geometric Constructions Special Topics Sylow Theorems Finite Simple Groups Generators and Relations Symmetry Groups Symmetry and Counting Cayley Digraphs of Groups All divisors of Use the corollary to Theorem 4.
Since K has two reflections it has a non-trivial rotation. The order of the subgroup of K consisting of rotations only must be 5 or 7 since it divides 35 and is proper. Thus, by Theorem 4. Clearly 0 is in H. If m and n are in H then m has the forms 8m1 and 10m2 and n has the forms 8n1 and 10n2. So, H is a subgroup of Z. If an infinite group had only a finite number of subgroups then it would have only a finite number of cyclic subgroups.
If each of these cyclic subgroups is finite then the group would be finite since every element is in the cyclic subgroup generated by itself. So the group contains at least one infinite cyclic subgroup, call it hai. Then the subgroups hai, ha2 i, ha3 i,. An infinite cyclic group does not have element of prime order. A finite cyclic group can have only one subgroup for each divisor of its order.
Another element of order p would give another subgroup of order p. D33 has 33 reflections each of which has order 2 and 33 rotations that form a cyclic group. To see this let H be the unique subgroup of G of order Then H also contains the unique subgroups of G orders 1, 3, and 5. The same argument works when 15 is replaced by any positive integer n. Then, by Theorem 4.
In general, if an Abelian group contains cyclic subgroups of order m and n where m and n are relatively prime, then it contains subgroups of order d for each divisor d of mn. In general, if an Abelian group contains cyclic subgroups of order m and n, then it contains subgroups of order d for each divisor d of the least common multiple of m and n.
Say a and b are distinct elements of order 2. If a and b commute, then ab is a third element of order 2. If a and b do not commute, then aba is a third element of order 2. So, by Theorem 4. Mimic Exercise From Theorem 4. If G is cyclic, then by Theorem 4. So, G is not cyclic. So, again G is not cyclic. Say b is a generator of the group. Since there are 3 choices for each of a, b, and c, the group has 27 elements. Noting that adding any polynomial in the group to itself 3 times results in a polynomial with coefficients 0 modulo 3, we see that the largest order of any element is 3 so G is not cyclic.
By Corollary 2 Theorem 4. Since reflections have order 2, any cyclic subgroup of order 4 must be generated by a rotation. So, by Theorem 4 there is exactly one cyclic subgroup of order 4.
Suppose that K is a subgroup of Dn of order 4 where n is odd. By Exercise 5. Since the subgroup of Dn of all rotations is cyclic of odd order, K cannot consist of 4 rotations nor have a rotation of order 2.
This rules out both cases for K. By Exercise 25 of Chapter 3 these are the only non-cyclic subgroups of order 4. First observe that the set of all rotations is the only cyclic subgroup of Dn of order n.
Rm , RF, R2 F,. G is a group because it is closed. It is not cyclic becasue every nonzero element has order 3. Since m and n are relatively prime, it suffices to show both m and n divide k. If both ab and ba have infinite order, we are done. By cancellation, ab cannot be a, a2 of b. If G has an element of order 4, G is cyclic. The remaining case is handled by Exercise 35 of Chapter 2. Let b be either of the remaining two elements of G.
We may now assume that every nonidentity element of G has order 2. Let c be the remaining element of G. Suppose that G is an Abelian group of order 6 but has no element of order 6. G cannot have an element a of order 5 for letting b denote the element not in hai we know that G would contain e, a, a2 , a3 , a4 , b and ab, which is too many.
Moreover, G has no element a of order 4 for then G would contain e, a, a2 , a3 , b, ab and a2 b, which is too many. So, we may assume that all nonidentity elements of G have orders 2 or 3. Now note that G cannot have two elements, say a and b, of order 2. So, G must have an element a of order 2 and an element b of order 3.
Then ab has order 6. The elements of finite order do not always form a subgroup in a non-Abelian group. See Exercise 8 of this set. If x commutes with a then a belongs to the subgroup C x. By closure, ak belongs to C x and therefore x and ak commute. Now suppose that x and ak commute. By the Corollary to Theorem 0. In D11 let a be a reflection and let b be a rotation other than the identity.
Solution from Mathematics Magazine. Let a be an arbitrary element of S. Thus S is a subgroup. Let H be the union. Since the square of each of these is 1 they are relatively prime to 4n. Let a and b belong to H. H contains the identity. First suppose that G is not cyclic. If T and U are not closed, then there are elements x and y in T and w and z in U such that xy is not in T and wz is not in U. So, there is an element b not in hai that also has order p.
Let G be the group of all polynomials with integer coefficients under addition. Let Hk be the subgroup of polynomials of degree at most k together with the zero polynomial the zero polynomial does not have a degree. Now that we know that elements from H commute it follows that ab divides lcm a , b and from this we have that H satisfies the subgroup test. This argument is valid for all positive integers.
By Exercise 15 of Chapter 4 K is a subgroup. Suppose that there is an element b such that b does not divide a. Then there is some prime-power divisor pn of b such that pn does not divide a. For the non-Abelian example, consider D3. Thus 1 is the identity it is obvious that the operation is commutative. Thus 2 generates all integers. By Theorem 5. We find the orders by looking at the possible products of disjoint cycle structures arranged by longest lengths left to right and denote an n-cycle by n.
So, for S6 , the possible orders are 1, 2, 3, 4, 5, 6; for A6 the possible orders are 1, 2, 3, 4, 5. We see from the cycle structure of S7 shown in Example 4 that in A7 the possible orders are 1, 2, 3, 4, 5, 6, 7. The mapping from Z to Z that takes x to 2x is one-to-one but not onto. These eight elements form a subgroup. The same process shows that when n is odd we get an even permutation.
An even number of two cycles followed by an even number of two cycles gives an even number of two cycles in all. So the Finite Subgroup Test is verified.
If all members of H are even we are done. Thus, there are at least as many even permutations are there are odd ones. Thus, there are at least as many odd permutations are there are even ones. By Exercise 23, either every element of H is even or half are even and half are odd. In the latter case, H would have even order. The identity is even; the set is not closed. An odd permutation of order 4 must be of the form a1 a2 a3 a4.
There are 6 choices for a1 , 5 for a4 , 4 for a3 , and 3 for a4. An even permutation of order 4 must be of the form a1 a2 a3 a4 a5 a6.
As before, there are 90 choices a1 a2 a3 a4. This gives elements of order 4 in S6. A permutation in S6 of order 2 has three possible disjoint cycle forms: a1 a2 , a1 a2 a3 a4 and a1 a2 a3 a4 a5 a6. This gives 45 distinct elements. For a1 a2 a3 a4 a5 a6 there are 6! This gives 15 elements. So, the total number of elements of order 2 is Any product of 3-cycles is even whereas is odd.
But that requires at least 9 symbols and we have only 7. Observe that if we start with a 9-cycle a1 a2 a3 a4 a5 a6 a7 a8 a9 and cube it we get a1 a4 a7 a2 a5 a8 a3 a6 a9. So, 3 and 2 divide m. Thus, we can say that m is a multiple of 6 but not a multiple of So, by Theorem 3. An element of order 5 in A6 must be a 5-cycle.
But the same 5-cycle can be written in 5 ways so we must divide by 5 to obtain Theorem 5. Repeated use of this argument finishes the proof.
Examining Sn for each n up to 8 as in Example 4 we see that Sn has no element of order greater than 2n. Note that the permutation is in both A8 and A10 and has order Cycle decomposition shows any nonidentity element of A5 is a 5-cycle, a 3-cycle, or a product of a pair of disjoint 2-cycles. One possibility is h i. The orders of the elements divide the order of the group. Any element from An is expressible as a product of an even number of 2-cycles. For each pair of 2-cycles there are two cases.
One is that they share an element in common ab ac and the other is that they are disjoint ab cd. Label"the positions ace to king as 1 through But S13 has no elements of order From the elements on the diagonal in Table 5.
Elements of A4 have order 3, 2 or 1. A similar argument applies to K. Observe that 56 belongs to A6 but is not in H. Now continue in this way. The product of an element from Z A4 of order 2 and an element of A4 of order 3 would have order 6.
The product of an element from Z A4 of order 3 and an element of A4 of order 2 would have order 6. But A4 has no element of order 6. This implies that the set is closed under multiplication and is therefore a group. In fact, 11 could be appended to every permutation written in cycle form making it both even and odd. An automorphism of a cyclic group must carry a generator to a generator. U 8 is not cyclic while U 10 is. D12 has an element of order 12 and S4 does not.
Taking the inverse of both sides proves that G is Abelian. By Theorem 6. See Theorem 0. Note that both H and K are isomorphic to the group of all permutations on the four symbols, which is isomorphic to S4. Observe that h2i, h3i,. This follows directly from the subgroup tests. Z60 contains cyclic subgroups of orders 12 and 20 and any cyclic group that has subgroups or orders 12 and 20 must be divisible by 12 and So, 60 is the smallest order of any cyclic group that subgroups isomorphic to Z12 and Z It is enough to prove that the mapping is one-to-one.
The operation preserving condition is Exercise 9 of Chapter 0. By Part 2 of Theorem 6. The inverse of a one-to-one function is one-to-one. This shows that Tg is a one-to-one function. U 20 has three elements of order 2 whereas U 24 has seven. This map preserves both addition and b a multiplication. First observe the Z is a cyclic group generated by 1.
By property 3 of Theorem 6. D4 has 5 elements of order 2, the quaternion group has only 1. For any a1 ,. Since G is finite, implies onto. This follows directly from property 3 of Theorem 6.
Assume a and b belongs to H. Assume that a and b belong to H. Multiplication is not preserved. By part 2 of Theorem 6. Also, by Part 2 of Theorem 6.
The argument given in Exercise 40 shows that an isomorphic image of Q has the form aQ where a is a nonzero rational. Consider first the cycle that begins with e. That cycle is e, eg, eg 2 ,. Continue in this fashion. It follows that the integer r maps to ar.
The cosets are H, 7H, 13H, 19H. They are ha5 i, aha5 i, a2 ha5 i, a3 ha5 i, a4 ha5 i. Let F and F 0 be distinct reflections in D3. The subgroup of order 1 is hei. By Corollary 4 of Theorem 7. Then since we know that g is a common divisor of both m and n. Let h be any element in H. Since G has odd order, no element can have order 2.
First suppose G is infinite. So we may assume all elements have finite order. Now use Theorem 4. Let H be the subgroup of order p and K be the subgroup of order q. The possible orders are 1, 3, 11, By the Corollary of Theorem 4.
The identity accounts for one more. So, at most we have accounted for 31 elements. So, a group with exactly 20 elements of order 11 must have exactly 34 elements of order 5. This contradicts the Corollary to Theorem 4. Since the reflections in a dihedral group have order 2, the generators of the subgroups of orders 12 and 20 must be rotations. The smallest rotation subgroup of a dihedral group that contains rotations of orders 12 and 20 must have order divisible by 12 and 20 and therefore must be a multiple of So, D60 is the smallest such dihedral group.
But in each of the last three cases Theorem 4. Let a have order 3 and b be an element of order 3 not in hai. See the Cayley Table for D3 on the inside back cover. Suppose G is a group of order Let a be any non-identity element in G. So, if any of these cases occur we are done. Thus we may assume that all 62 non-identity elements in G have order 7. But by the Corollary to Theorem 4.
Then by Theorem 7. Since reflections have order 2 the subgroup must consist entirely of rotations and the subgroup of all rotations is cyclic. Similarly, if a has order 6 or 4 then there is an element of order 2.
So, we may assume that all 11 nonidentity elements have order 3. Since this is a contradiction, one of the earlier cases must occur. Suppose G is a group with distinct subgroups hai and hbi of order 5.
Because 5 is prime, the identity is the only element common to the two subgroups groups. Suppose that H is a subgroup of A5 of order We claim that H contains all 20 elements of A5 that have order 3. The same argument shows that H must contain all 24 elements of order 5. We claim that H contains all 24 elements of A5 that have order 5.
An analogous argument shows that A5 has no subgroup of order Since e is in H, K is not empty. Suppose that H is a subgroup of S5 of order But any subgroup generated by an element of order 2 and an element of order 3 that commute has order 6. This contradicts the fact shown in Example 5 that A4 has no subgroup of order 6. The circle passing through Q, with center at P.
It is the set of all permutations that carry face 2 to face 1. Since the order of G is divisible by both 10 and 25 it must be divisible by But the only number less than that is divisible by 50 is Closure and associativity in the product follows from the closure and associativity in each component.
The identity in the product is the n-tuple with the identity in each component. The inverse of g1 , g2 ,. Every nonidentity element in the group has order 2. Each of these generates a distinct subgroup of order 2. A corresponding statement holds for the external direct product of any number of groups. In general, the external direct product of any number of groups is isomorphic to the external direct product of any rearrangement of those groups.
See also Theorem 8. Yes, By Theorem 8. Z9 has 6 elements of order 9 the members of U 9. Any of these together with any element of Z3 gives an ordered pair whose order is 9.
This gives us a total of In the general case observe that by Theorem 4. The first two are Abelian and the second two are not. D6 has an element of order 6 and A4 does not. The group of rotations is Abelian and a group of order 2 is Abelian; now use Exercise 4. In general, if the external direct product of any number of groups is cyclic, each of the factors is cyclic. Since any cycle group of even order has exactly 1 element of order 2 and 1 of order 1 there are only 3 choices for a, b.
But isomorphisms preserve order. If exactly one ni is even then x is the unique element of order 2. Otherwise x is the identity. Each cyclic subgroup of order 6 has two elements of order 6. So, the 24 elements of order 6 yield 12 cyclic subgroups of order 6. In general, if a group has 2n elements of order 6 it has n cyclic subgroups of order 6. Recall from the Corollary of Theorem 4. S3 In each position we must have an element of order 1 or 2 except for the case that every position has the identity.
For the second question, we must use the identity in every position for which the order of the group is odd. Using the fact that an isomorphism from Z12 is determined by the image of 1 and the fact that a generator must map to a generator, we determine that there are 4 isomorphisms. Since 2, 0 has order 2, it must map to an element in Z12 of order 2. The only such element in Z12 is 6. Since 1, 0 has order 4, it must map to an element in Z12 of order 4.
The only such elements in Z12 is 3 and 9. A second subgroup of order 5 is ha2 i where a2 is any element not in ha1 i. A third subgroup of order 5 is ha3 i where a3 is any element not in ha1 i or ha2 i. This gives us Z3 and Z4. As before no prime can be greater than 3. Identify A with 0,0 , T with 1,1 , G with 1,0 and C with 0,1. Then a string of length n of the four bases is represented by a string of 0s and 1s of length 2n and the complementary string of a1 a2.
So we have 2 choices for each of a, b, c, and d. This gives 16 in all. Now use Theorem 8. So, all cubes in U 55 are distinct.
We need to find relatively prime m and n so that both U m and U n are divisible by 5. U This is the same argument as in Exercise 77 with 5 replaced by 3. Um n is a subgroup of Uk n. Obviously, 72 is one choice. Step 3 of the Sender part of the algorithm fails. So, we need to compute mod The result is , which converts to NO. Consider the finite and infinite cases separately.
For the infinite case, use Exercise 2 of Chapter 6. See Theorem 6. When K is characteristic, and automorphism of G is an automorphism of K, which in turn, is an automorphism of N when N is characteristic. All nonidentity elements of G and H have order 3.
Let a and b be distinct nonidentity elements in G of order 2. Note that H1 is a subgroup of G and the product of all its elements is e. If not, then let c be an element of G not in H1. If not, then let d be an element of G not in H2.
Continuing in this way finishes the proof. Let x and y belong to H. Observe that the exponent of a finite group is the least common multiple of the order of all the elements of the group.
For U n to have exponent 2 every nonidentity element must have order 2. So every element can be expressed in the desired form. But there is no rational number whose cube is 2. By property 2 of Theorem 6. In R under addition every nonzero element has infinite order. Fix a1 ,. Then x1 ,. Count elements of order 2. See Exercises A pyramid with a regular pentagonal base.
There are 7! This makes in all. Notice the second coordinate is triple the first minus 1. These steps are reversible. So, there is a prime p that divides both a and b.
Use the previous exercise. First observe that by Theorem 8. By Theorem 8. It follows from Corollary 4 of Theorem 7. D6 has an element of order 6 but S4 does not. Also, 1n So, by induction, 12 and This means that every 2-cycle not involving n can be generated.
H contains the identity so H is not empty. This proves that H is a subgroup. By Theorem 9. Since H and K have order 2 they are both isomorphic to Z2 and therefore isomorphic to each other. This means that H has at least 27 elements. First observe that every proper subgroup of D4 is Abelian. Now use Theorem 9. If h or k is infinite, so is g. Suppose H is any subgroup of index 2. Now use Corollary 2 to Theorem 4. Suppose that H is a proper subgroup of Q of index n. The proof is valid for any integer.
G and the trivial subgroup are normal. By Exercise 9 of this chapter By Example 14 of Chapter 3, Z D13 is the identity. Then by Theorem 9. Say aH has finite order n. But this implies that an and therefore a is finite. Say H has an index n. Use part a of this exercise and part a of Exercise 4 of Supplementary Exercises for Chapters So, K is not normal in D4.
The examples elucidate the definitions, theorems, and proof techniques. The exercises facilitate understanding, provide insight, and develop the ability of the students to do proofs. There is a large number of exercises ranging from straight forward to difficult and enough at each level so that instructors have plenty to choose from that are most appropriate for their students.
The exercises often foreshadow definitions, concepts, and theorems to come. Many exercises focus on special cases and ask the reader to generalize. Generalizing is a skill that students should develop but rarely do. Even if an instructor chooses not to spend class time on the applications in the book, I feel that having them there demonstrates to students the utility of the theory.
Changes for the ninth edition include new exercises, new examples, new biographies, new quotes, new appliactions, and a freshening of the historical notes and biographies from the 8th edition.
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