Suppose that G is cyclic and G = (a) where |a| = 20. How many subgroups does G have? 2 5 4 CO

Q: Show that the cyclic group of n objects, Cn, may be represented by r", m = 0, 1, 2,...,n– 1. Here r…

A: Given  Cn is a cyclic group of n elements To show Cn is generated by some element r of Cn. That is…

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

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

Q: (d) Is a21 a generator of G? Why or why not? (e) Let H < G. If |H| = 7, then which cyclic subgroup…

A:

Q: 3. Consider the group S. a) For the subgroup H= {(1),(13)} , write all cosets that can be formed. b)…

A:

Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

A:

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

A:

Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…

A: Q1.  Third option is correct. Q2. Second option is correct.

Q: C. Find all subgroups of the group Z12, and draw the subgroup diagram for the subgroups.

A:

Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

Q: Suppose that X and Y are subgroups of G if |X|-32 and IYI=48, then what is the best possible of Xn…

A:

Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: Q2)) prove that the center of a group (G, ) is a subgroup of G and find the cent(H) where H = (0, 3,…

A:

Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

A:

Q: a) Make a b) Show that if with |K| list of all subgroups of Q8 K is is a subgroup of Qs. k is…

A:

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: Suppose that 0:G G is a group homomorphism. Show that () o(e) = 0(e) (i) For every gEG, ($(g))¯1…

A:

Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…

A:

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

Q: (d) Is a21 a generator of G? Why or why not? (e) Let H < G. If |H| = 7, then which cyclic subgroup…

A:

Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

A:

Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

A:

Q: How many cyclic subgroups does have U(15) have? 4 3

A: We will determine the cyclic subgroup generated by each element of G

Q: 3. (a) List the subgroups of Z15 (b) List the subgroup of Z24 4 TEa cyclic onoun Tof G is normal in…

A: Note: " As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: How many subgroups does Z20 have? List a generator for each ofthese subgroups. Suppose that G =…

A:

Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

A:

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

A:

Q: Question 2. Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G…

A: Given: G is a group that has exactly one non-trivial proper subgroup. To prove: G is cyclic and…

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: Suppose that $:G→Gis a group homomorphism. Show that ) p(e) = ¢(e') iI) For every gEG, ($(g))-1 =…

A: Definition of Homomorphism: Let (G,∘)  and (G',*)be two groups. A mapping ϕ:G→G'is said to be…

Q: (a) Let G be any group. Let H <G and K < G be subgroups with |H|= |K| = p, where %3D is prime. Show…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: 5. Let G be a group and n e Z+ be fixed. Show that H = {a" | a € G} is a subgroup of G

A:

Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…

A:

Q: Find all cosets of the subgroup of Z12.

A:

Q: Q/ How many non-trivial subgroups in s, ?

A: For the given statement

Q: 5. Consider the "clock arithmetic" group (Z15,0) a) Using Lagrange's Theorem, state all possible…

A:

Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…

A: We have to prove given result:

Q: Suppose that 0: G G is a group homomorphism. Show that 0 $(e) = ¢(e') (1) For every gEG,…

A:

Q: 43. Consider the subgroup H = {0,4} of the %3D group G = (Zg, +8, -8). Find the right cosets of H in…

A: G = (Z8, +8, •8) and H ={0, 4} be subgroup of G

Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is

A: It is given that H and K are subgroups of G and H=16, K=28. Since H and K are subgroups of G, H∩K≤H…

Q: List all the elements of the cyclic subgroup of U(15) generated by 8. 2. Which of the following…

A: We have to find the all elements of cyclic subgroup of U(15) generated by 8.

Q: For A, the alternating subgroup of S, show that it is a normal subgroup, write out the cosests, then…

A:

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

A:

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…

A:

Q: Lemma 5 Let G be a group and Ha subgroup of G. Prove that the normalizer, Nc(H), is a subgroup of G…

A:

Q: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

A:

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

Q: efine xHx-1= {xhxh Hx is a subgroup of G. His cyclic, then xHx E H is cyc

A: Given: G and H be group and subgroup. xHx-1=xhx-1|h∈H

Q: Suppose that | G|= 20. What're possible orders for a subgroup of G. 35.

A: We can find the subgroup of G as below.