Group Theory Exercises

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) Prove that there is a mapping from a set to itself that is one-to-one but not onto if and only if there is a mapping from the set to itself that is onto but not one-to-one.

(2) Show there does not exist a one-to-one mapping from onto and then show there does not exist a mapping from onto

(3) Let and be sets with Show that the mapping is one-to-one if is one-to-one. Show that the mapping is onto if is onto.

(4) Let and be sets with Show that the mapping is one-to-one if is onto. Show that the mapping is onto if is one-to-one.

(5) State and prove the Universal Mapping Property.

(6) Prove that the set of automorphisms of a vector space, with multiplication given by composition of maps, is a group.

(7) Show that the set of all bijections from to denoted by forms a group under composition of mappings.

(8) Let be an injection from to and let be a known element of Show that there is at least one mapping from to such that is the identity element in and that must be a surjection.

(9) Let be a surjection from to Using the Axiom of Choice, show that there is at least one injective mapping from to such that is the identity of

(10) Prove that concatenation of input strings for a Finite-State Machine is an associative operation. Does this operation form a group?

(11) Give a state diagram for a machine with input alphabet that can be used to determine if the input string contains exactly (a) two and (b) three

(12) Show that the external direct product of a finite set of groups is itself a group.

(13) Define a group operation on the set of points on a non-singular elliptic curve with Weierstrass equation where and are real numbers.

(14) Suppose is a group and is a one-to-one mapping from onto the set Prove that defining a multiplication on by makes into a group under

(15) If then is not a group with respect to composition.

(16) Show that is a group using composition as an operation.

(17) Show that is a group using composition as an operation.

(18) Give three more examples of groups using composition as an operation.

(19) If is a finite group of even order, then contains an element such that

(20) Let be a semigroup. Then is a group if and only if all the equations and have solutions in

(21) Let be an equivalence relation on a monoid such that and imply for all Then the set of all equivalence classes of under is a monoid under the binary operation defined by where denotes the equivalence class of If is an (Abelian) group, then so is

(22) Prove that the following relation of the additive group of rational numbers is an equivalence relation : if and only if Prove that the set of equivalence classes is an infinite Abelian group which is sometimes called the group of rationals modulo one.

(23) Let be a prime and let be the following subset of the group :

Show that is an infinite group under the addition operation of

Prove the following statements.

(24) Every permutation can be written as a cycle or as a product of disjoint cycles.

(25) If the pair of cycles and have no entries in common, then

(26) The order of a permutation written in disjoint cycle forms is the least common multiple of the lengths of the cycles.

(27) Every permutation in is a product of

(28) If a permutation can be written as a product of an even number of -cycles, then every decomposition of into a product of -cycles must have an even number of -cycles.

(29) The set of even permutations in forms a subgroup of

(30) Let be a group of permutations on a set Let and define the stabilizer of in to be the subset of , Prove that is a subgroup of

(31) Prove the following statement:Let be a non-empty subset of a group and define a relation on by if and only if Show that is an equivalence relation if and only if is a subgroup of

(32) Prove the following statement: If is a group and is a non-empty family of subgroups, then is a subgroup of

(33) Prove the following statement: If is a group and is a non-empty subset of then the subgroup generated by consists of all finite products where and for every In particular for every

(34) Let be a group and a family of subgroups. State and prove a condition that will imply that is a subgroup, that is, that Give an example of a group and a family of subgroups such that

(35) Determine the group of symmetries of a regular cube.

(36) Determine the group of symmetries of a pyramid with a square base.

(37) Determine the group of symmetries of a regular tetrahedron.

(38) Prove that the reflexive, symmetric, and transitive properties of an equivalence relation are independent..

(39) Prove that if is a non-empty set, then the assignment defines a bijection from the set of all equivalence relations on onto the set of all partitions of

(40) Prove that if is any function, and is the equivalence relation defined on by letting if for all then there is a one-to-one correspondence between the elements of the image of under and the equivalence classes of the relation Show the factorization in a diagram.

(41) Let Define on by if there exists a nonzero real number such that and Show that is an equivalence relation on and give a geometric description of the equivalence class of

(42) Let where is given in exercise 13. Show that is a well-defined subset of and what are the subsets called? Also show that any two points in determines a unique subset with the same form as

(43) If is a cyclic group and is any group, then every homomorphism is completely determined by the element

(44) For each prime the additive subgroup of is generated by the set

(45) Prove that the set of all subgroups of a group , partially ordered by set theoretic inclusion, forms a complete lattice in which the greatest lower bound of is and the least upper bound is

(46) Give an example of groups such that and no is isomorphic to any

(47) Let be an additive Abelian group with subgroups and Show that if and only if there are homomorphisms such that and where is the map sending every element onto the zero (identity) element, and for all

(48) Let be finite cyclic groups. Then is cyclic if and only if

(49) Let be nontrivial normal subgroups of a group and suppose Prove that is in the center of or intersects one of or nontrivially. Give examples to show that both possibilities can occur when is non-Abelian.

(50) Prove that the following conditions on a finite group are equivalent: (i) is prime (ii) and has no proper subgroups, (iii) for some prime

(51) Let be subgroups of a group Prove that is a subgroup of if and only if

(52) If and are subgroups of finite index of a group such that and are relatively prime, then

(53) Prove that if and are subgroups of a group such that then

(54) Let be subgroups of a group such that and Show that

(55) Show that if is a group of order then contains an element of order Also show that if is odd and Abelian, there is only one element of order 2.

(56) If and are subgroups of a group then

(57) Show that if are primes, a group of order has at most one subgroup of order

(58) Let be the group (under ordinary matrix multiplication) generated by the complex matrices and where Show that is a non-Abelian group of order 8. What is called?

(59) Let be a group and such that (i) , (ii) , (iii) (iv) (v) Show that and

(60) Let be an Abelian group containing elements and of orders and respectively. Show that contains an element whose order is the least common multiple of and

(61) If is a homomorphism, and has finite order in then is infinite or divides

(62) Let be the multiplicative group of all non-singular matrices with rational entries. Show that has order 4 and has order 3, but has infinite order. Conversely, show that the additive group contains nonzero elements of infinite order such that has finite order.