Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects = {e}.

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: Consider the groups A = Z3 © Z9 and B = Z27. Prove that these two groups are not isomorphic.

A: The two groups G and G' are isomorphic iff there exists an onto isomorphism between them .

Q: If H and K are subgroups of G and g belongs to G, show that g(H N K) = gH n gK.

A: Given: H and K are subgroups of G and g.To prove: gH∩K=gH∩gK

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

A:

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

A:

Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let  be  a  group  and  assume  that Suppose  that Since,  and  is  a  group, it follows  that…

Q: let x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: Calculate G/H for G = V, the Klein’s four-group and H = (b).

A: The Klein four-group is defined by the group presentation V=a, b| a2=b2=ab=c2=e Given: G=e, a, b, c…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

A:

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: B A F (Ε

A:

Q: = n and o(b) = m Let G be a group and suppose a, b E G satisfy ab = ba. Assume that o(a) : with (n,…

A: Given G is a group, such that a, b∈G satisfy ab=ba. Also, we have oa=n and ob=m with n, m=1.   To…

Q: Help: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be…

A: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be a…

Q: Decide if each of the following sets H is a subgroup or not of the group G = R XR, where the group…

A:

Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…

Q: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: 1. Prove or disprove that each of the cises 52 of Section 3.1 is cyclic. a. Z2 × Z3

A:

Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

A: