Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same for the group ((Z/10Z^, *).

Q: The elements of the quotient group (/18) are 1- 2- {.-208,-108,008,108,208,.} 3- (0,1,2,3, .8) 4-…

A:

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

A:

Q: for each of the group Z2,Z4,Z6 INDICATE WHICH CYCLIC. FOR those cyclic list  all the generators?

A:

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

A:

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: Define Group theory ?

A: To define group theory

Q: Express U(165) as an internal direct product of proper subgroups infour different ways

A:

Q: How many elements of a cyclic group with order 10 have order dividing 5?

A:

Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

A:

Q: Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external…

A: We know the following theorem. Theorem: Aut(Zn) is isomorphic to the group of multiplicative units…

Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: How many nonisomorphic abelian groups of order 80000 are there?

A:

Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

A:

Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

A:

Q: In practice, what can be the differences between the relationships of two pairs of groups, knowing…

A: Homomorphism -  It is  algebraic term for a function having some algebraic operations. For a group…

Q: Does the Z base 5 form a cyclic group w.r.t. addition modulo 5? Justify. If yes, find all the…

A:

Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

Q: Determine all cyclic groups that have exactly two generators.

A:

Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: How many elements of a cyclic group with order 14 have order 7?

A:

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

A:

Q: List the identity element and each other element (along with their inverses) for the group U(18)

A: we have to list the identity element and each other element (along with their inverses) for the…

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Which abelian somorphic to groups subyraups of Sc. Explin. are

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

A:

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

A:

Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

A:

Q: EQ6/ Give example of a group such that has element of order 7 and nurmal subgruup of Order 3.

A: A detailed solution is given below.

Q: Find the order of the group and the order of each element in the group. In each case, how are the…

A: The no. of elements present in a group is the order of the group. n is the least positive integer…

Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

A:

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

A:

Q: The group (Z6,6) contains only 4 subgroups

A:

Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups  are cyclic groups. A group G is cyclic if G=<g> for some g in G, where…

Q: What limits will include the middle 80% of the group?

A: The required calculation can be defined by the formula Z=x-μσP(a≤Z≤b)=P(b≤Z)-P(a≤Z) Given, What…

Q: The group (Z6,+6) contains only 4 subgroups

A:

Q: Since 11 is an element of the group U(100); it generates a cyclic subgroup Given that 11 has order…

A:

Q: True or false? Every group of 125 elements has at least 5 elements that commute with every element…

A: Let G be a group whose order is 125 ⇒G=125=53 Center of a group G ( ZG ) is the set of all those…

Q: t subgroups and quotient groups of a solvable group are solvable.

A:

Q: Represent the symmetry group of an equilateral triangle as a groupof permutations of its vertices

A: To determine: The symmetry group of an equilateral triangle as a  group of permutations of its…

Q: What is the numbers group of

A: The given number is 5. The value of 5 is 2.23606...

Q: a) Clearly state all the subgroups of the additive group Explain why/how you have found all the…

A: We have to write subgroup of z_9.