1 Fundamentals 2 The Integers 3 Groups 4 More On Groups 5 Rings, Integral Domains, And Fields 6 More On Rings 7 Real And Complex Numbers 8 Polynomials expand_more
4.1 Finite Permutation Groups 4.2 Cayley’s Theorem 4.3 Permutation Groups In Science And Art (optional) 4.4 Cosets Of A Subgroup 4.5 Normal Subgroups 4.6 Quotient Groups 4.7 Direct Sums (optional) 4.8 Some Results On Finite Abelian Groups (optional) expand_more
Problem 1TFE: True or False Label each of the following statements as either true or false. aHHa where H is any... Problem 2TFE: True or False
Label each of the following statements as either true or false.
2. Let be any subgroup... Problem 3TFE: True or False Label each of the following statements as either true or false. Let H be any subgroup... Problem 4TFE: True or False
Label each of the following statements as either true or false.
4. The elements of... Problem 5TFE: True or False Label each of the following statements as either true or false. The order of an... Problem 6TFE: True or False
Label each of the following statements as either true or false.
The order of any... Problem 7TFE: True or False Label each of the following statements as either true or false. Let H be a subgroup of... Problem 8TFE: True or False
Label each of the following statements as either true or false.
8. Every left coset of... Problem 1E: 1. Consider , the groups of units in under multiplication. For each of the following subgroups in ,... Problem 2E: For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H... Problem 3E: In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its... Problem 4E: In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication... Problem 5E: Let H be the subgroup (1),(1,2) of S3. Find the distinct left cosets of H in S3, write out their... Problem 6E: Let be the subgroup of .
Find the distinct left cosets of in , write out their elements,... Problem 7E: In Exercises 7 and 8, let be the multiplicative group of permutation matrices in Example 6 of... Problem 8E Problem 9E: Let be a subgroup of a group with . Prove that if and only if
Problem 10E: Let be a subgroup of a group with . Prove that if and only if .
Problem 11E: Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?
Problem 12E: Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order... Problem 13E: Let H be a subgroup of the group G. Prove that if two right cosets Ha and Hb are not disjoint, then... Problem 14E: Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is... Problem 15E Problem 16E: Let H be a subgroup of the group G. Prove that the index of H in G is the number of distinct right... Problem 17E: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba. Problem 18E: Let G be a group of finite order n. Prove that an=e for all a in G. Problem 19E: Find the order of each of the following elements in the multiplicative group of units .
for
for
... Problem 20E: Find all subgroups of the octic group D4. Problem 21E Problem 22E: Lagranges Theorem states that the order of a subgroup of a finite group must divide the order of the... Problem 23E: Find all subgroups of the quaternion group. Problem 24E: Find two groups of order 6 that are not isomorphic. Problem 25E: If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G. Problem 26E: Let p be prime and G the multiplicative group of units Up=[a]Zp[a][0]. Use Lagranges Theorem in G to... Problem 27E: Prove that any group with prime order is cyclic. Problem 28E: Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is... Problem 29E: Let be a group of order , where and are distinct prime integers. If has only one subgroup of... Problem 30E: Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G... Problem 31E: A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements... Problem 32E: (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup... format_list_bulleted
