Suppose n km for positive integers k, m. In the additive group Z/nZ, prove that |[k],| = m, where [k]n = nZ + k E Z/nZ the right coset of k.

Q: Suppose F1 and F2 are distinct reflections in a dihedral group Dn.Prove that F1F2 ≠ R0.

A: Given: F1 and F2 are distinct reflections in a dihedral group Dn. To Prove: F1F2≠R0.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Let Sn denote the symmetric group on n letters. Answer the following: h) Let n = 7. Find all values…

A: Sn denote the symmetric group on n letters Given n = 7. We have to find all values of k such that…

Q: Find the right cosets of the subgroup H in G for H = ((1,1)) in Z2 × Z4.

A: Let, the operation is being operated with respect to dot product. Elements of ℤ2=0,1 Elements of…

Q: Let a be an element of a group G such that o(a) = r. Let m be a positive integer such that (m,r) =…

A: a∈GOa=rgcdm,r=1 We have to prove Oam=r

Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

A:

Q: Let H = {z E C* | Izl = 1}. Prove that C*/H is isomorphic to R+, the group of positive real numbers…

A:

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: Let F be a field, and consider the group G := 1 a b {6 {(₁; ;) = 01 с : a, b, c E F ≤ GL3 (F). : a,…

A: Center of group is set of all elements which commute with all elements of G

Q: Exercise 7.19. Let G be a finite group, H ≤ G, |H| = 2k + 1 for some non-negative integers k, and…

A:

Q: answer this three question thx！

A: Q1.The value of X:132024y+4​ where y∈Z.Q2. The table is given in the attached paper.Q3. 1)No, The…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

A:

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

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: 1, and let ø denote Let m and n be positive integers with gcd(m, n) Euler's o-function (Definition…

A: Given: Let's determine the following a) Let's verify that (a,b) is a generator for the group ℤnℤ×ℤmℤ…

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…