What is the largest order among the orders of all the cyclic subgroups of Z6 x Z8? of Z12 x Z15?

Q: Show that the group of positive rational numbers under multiplicationis not cyclic. Why does this…

A: To Show: A group of positive rational numbers under multiplication is not cyclic and why does this…

Q: What is the minimum number of generators needed for Z2 ⨁ Z2 ⨁ Z2?Find a set of generators and…

A: Given: Group, Z2⊕Z2⊕Z2To find: The minimum number of generators and a set of generators and…

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

Q: Imagine  Z9* a cyclic group. What is order of an element a=2 and show the elements? Show all your…

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Q: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: give the order of the element in the in the factor group (2,5) + in (Z3 x Z6)/

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Q: A) (i), (ii) B C D E All (i), (iii) (ii), (iii) All Consider the subgroup 10Z of 2Z, which of the…

A: Elements of the group 2Z are {. . . . . ,-4, -2, 0, 2, 4, 6, 8, 10, . . . . } So, cosets of 2Z/10Z…

Q: Find cyclic subgroups of S4 that have three different orders.

A: There are more subgroups than just the cyclic ones.   Trivial:  there is <e> = {e}.   There…

Q: In Z, find all generators of the subgroup <10>  <12>.

A: The objective of the question is to find all generators of the subgroup in the set of integers…

Q: Show that a group G of order pqr  for distinct primes, p,q and r, is  solvalbe.

A: |G| = pqr.   Without loss of generality, let p<q<r. We claim two things :  (i) There exists a…

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Find all subgroups of U(7)

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Q: Show that there is no simple group of order pqr, where p, q, and rare primes ( p, q, and r need not…

A: We know the definition of simple group, A simple group is a group whose only normal subgroups are…

Q: Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the…

A: (i) Closed: Yes, the set of even integers is closed under addition, meaning that when two even…

Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

Q: List all the subgroups of Z, including the “obvious" or “trivial" ones. Explain how you generated…

A: Group: A group is a set G together with a binary operation a, b↦a∗b:G×G→G satisfying the following…

Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

A: Since , we have to find order of the factor group U(16)/<9> Concept: Order of factor group…

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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Q: Give a list of all groups of order 8 and show why they are not isomorphic. for this you can show…

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Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

Q: Let G = Z4 x Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the…

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Q: What do the Sylow theorems tell you about any group of order 100?

A: Let G be a group of order 100.

Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

Q: Find S4/N elements and show that they can be combined into a group. N={e,(23),(234),(243),(12)(34),…

A: We know,      | S4 | = 4! = 24     | N | = 12   So, | S4/N | = | S4 |/| N |                      =…