et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an element of order d

Q: Let G be the symmetry group of a circle. Explain why G containsDn for all n.

A: Given: G is the symmetry group of a circle.We need to explain that G contains Dn for all n.

Q: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

A:

Q: Find the order of each of the following elements in the multiplicative group of units Up. a. [2] for…

A: Consider the multiplicative group U13. Find the order of element 2.   The identity of a…

Q: Let G be a group and let a be a fixed element of G. Show that H = {x E G |xa = ax} is a subgroup of…

A:

Q: Let f be the following permutation: 1 2 3 4 5 6 7 f = 4 3 6 2 1 5 7 a. Determine the group to which…

A: Given permutation is f=12345674362157 f is permutation of 7 elements so it is belong to s7. i.e. sn…

Q: Let G be a finite group of order n . Let g be an element of G with order k. Prove that k|n

A: To prove ''Let G be a finite group of order n . Let g be an element of G with order k. Prove that…

Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin…

A:

Q: Consider the group G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4.  We have to list the elements of A4 in cycle notation, and we have to…

Q: Let G be a group of order n. For what value of n is the group G NOT cyclic?

A:

Q: Let G be the set of nonnegative integers. Is a group, |a - b| for all a, b, EG? where a * b =

A: We will check whether it is group or not.

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

A:

Q: Let K be a group with 8 elements. Let H be a subgroup of K and H<K. It is known that the order of H…

A: Given:- K is a group with 8 elements , H is a subgroup of K and H&lt;K. H has order of atleast 3.…

Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

A:

Q: Exercise 4.2. Let G be a group and a, b = G. Suppose that ab has finite order n. Prove or disprove…

A: Step 1: Step 2: Step 3: Step 4:

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

A:

Q: Let G be a group and let x ∈ G. Prove that if a has order of 12, then x3 has order 4.

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: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

A:

Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

A:

Q: Let G be a group under multiplication, G be a group under addition and o be an isomorphism from G to…

A:

Q: Explain why

A: We have to explain the marked step .

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

A:

Q: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

A:

Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

A:

Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…

Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

A:

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

A:

Q: Let G be a group with operation * and e be the identity element of G. Show that for any element a in…

A:

Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

A:

Q: Let G be a group and a 2 G. Then the set = {a" : ne Z}is a subgroup of G. 48.

A:

Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

A:

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

A:

Q: Let G be a group. It is known that a ∈ G, b ∈ G, |a| = 8, |b| = 15. Show that ∩ = {e} . where e…

A: Given G be a group and a∈G and b∈G such that a=8 and b=15 we have to show that a∩b=e Since, a=8…

Q: let n be a fixed natural number.Verify that the set  n.Z={n.k l kEZ} is a group of (Z,+).

A:

Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________

Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

A: