I need this Answer at half time Quickly. Please 5- *5) Let H and K be subgroups of a group G theno(H).0 (K)a) 0(H * K)2,if (O(H) > JO(G) and 0(K) > JO(G).O(HNK)b) 0(H * K) = 0(H).0(K),if H n K = {e}.c) All above.aO b

Given that H and K are subgroups of a group G.

Answered: 5- * 5) Let H and K be subgroups of a…

5- * 5) Let H and K be subgroups of a group G then о (н).0 (к) a) O(H * K) ,if (o(H) > Jo(G) and 0(K) > JO(G)). O(HNK) b) 0(H * K) = 0(H).0(K),if H n K = {e}. c) All above. O a O b O c

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

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...

See similar textbooks

Related questions

Q: 50

A: From the given information, it is needed to prove or disprove that H is a subgroup of Z:

Q: 2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

Q: abouis 13. Let (G, *) be cyclic group of finite order n and let a € G. Prove that ak is a generator…

Q: IL Show thai al subgroup of erder 125. 18. Show that there are no simple groups of order 255 =…

A: By using syllow theorems the solution is given as follows :

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: 10.3.3. Let G Z/4Z x Z/6Z, and let H be the subgroup of G generated by (2, 2). (a) What are the…

A: Given that G=ℤ/4ℤ×ℤ/6ℤ.i.e. G=ℤ4×ℤ6H is a subgroup of G generated by 2,2i.e. H=<2,2> oG=4×6=24

Q: let G be group.HiksGG k= Some 9be G. That is, H and k ase Cyolic Subgsoup of G. Does this imply…

A: Let G be any group. Let H and K are cyclic subgroup of G such that the for any a,b∈G, H=aK=b To…

Q: 6.10 Find all subgroups of Z,XZ4. 6.11 Find all subgroups of Z,xZ,>

A: To find All subgroups of z2*z4

Q: 3.40. Let G be an abelian group and n e N. Let H = {a e G : a" = e} and K = {a" : a e G}. Show that…

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

Q: 3. Find all Abelian groups (up to isomorphism) of order 100. From these isomorphism classes…

Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

Q: 6. For a positive integer n 2 4 and a prime number psn, let U, denote the union of p,n all p-Sylow…

Q: 3. Define Lie group.

Q: Please help me understand the following question and please explain the steps. Picture is below

Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e also x6=e therefore…

Q: 4) Let H and K be a subgroup of a group Gif HAK,KAG and HAG then HnK is not normal subgroup of G.…

A: Given, H and K are subgroups of a group G. Also, given that H is normal in G and K is normal in G.…

Q: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

Q: 32) Prove that every subgroup of Q8 in normal. For each subgroup, find the isomorphism type of its…

A: The elements of the group are given by, Q8=1,-1, i, -i, j,-j, k, -k Note that every element of Q8…

Q: se A cyclic group has a unique generator. f G and G' are both groups then G' nG is a group. A cyclic…

A: 1.False Thus a cyclic group may have more than one generator. However, not all elements of G need be…

Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

Q: (3) Suppose n= |T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic…

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group We have to find the order of the group and also check, This is a…

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 3x. Then, Ker(Ø) = * (0, 5, 10} None of the…

Q: Let Ø:Z50→Z15 be a group homomorphism with Ø(x)=7x. Then, Ker(Ø)= * O {0, 10, 20, 30, 40} None of…

Q: Theorem group tren G'45. Eremple Treorem is i0 9 N a group then Example abeliar.

Q: is] Let G be a group, and let Z(G) = {aeG; ag3ga for all geG}. Show that Z(G) is a subgroup of G. ]…

Q: Thus option (c) is correct. Let G be a group with G| = 96. Suppose H and K are subgroups of G with…

A: for abelian group -: condition -: 1 closure ,2 associative , 3 -: identity , 4 -: inverse ,5-:…

Q: Match each entry in the first column with the corresponding entry in the second column. (Multiple…

A: We use the results of cyclic group to give the answers.

Q: (8) Let n > 2 be an even integer. Show that Dn has at least n/2 subgroups isomorphic to the Klein…

Q: your a) G = (Z12,O); H = {0,2,4, 6, 8, 10} %3D

Q: Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G. Generalize to any group of…

A: The Result to be proved is: If G is a group of order p2, where p is a prime, then either G is cyclic…

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

Q: n 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

Q: True or False: (a) Two finite non-cyclic groups are isomorphic if they have the same order. (b)Let o…

Q: 4) If center of G equal G (ie cent(G) = G) then G is not necessary abelian group. true O false

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: 3. Show that Q has no subgroup isomorphic to Z2 × Z2.

A: The objective is to show that ℚ has no subgroup isomorphic to ℤ2×ℤ2

Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: 8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer

A: Note: since you have posted multiple questions . As per our guidelines we are supposed to solve one…

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 7x. Then, Ø-'(7) : O {0, 15, 30, 45} O {1, 6,…

Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

Q: An is a * normal subgroup of Sn not a subgroup of Sn subgroup of Sn but not normal None of the…

A: The solution is given as

Q: 2. (a) Let be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be…

Q: Let 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

Q: 6.17 Corollary If a is a generator of a finite cyclic group G of order n, then the other generators…

A: here we can use the given corollary to explain this example and also need to consider that which is…

Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

I need this Answer at half time Quickly. Please

5- *
5) Let H and K be subgroups of a group G then
o(H).0 (K)
a) 0(H * K)
2,if (O(H) > JO(G) and 0(K) > JO(G).
O(HNK)
b) 0(H * K) = 0(H).0(K),if H n K = {e}.
c) All above.
a
O b

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Single Variable$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
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 of K is 3, what are all the possible orders of H?
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
Find two groups of order 6 that are not isomorphic.
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
3. Consider the group under addition. List all the elements of the subgroup, and state its order.
Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.