Determine whether the following subsets (M, N and P) are subgroups of thestated groups.1i. The set M ={2|₁€Z} as a subset of the group (Q*, ×).2nii. The set N of n x n matrices with every diagonal entry equal to zero as asubset of the group GL₁(R).iii. Suppose that 0: G → Hand: G H are homomorphisms for the groups.G and H. Let P be the subset of the group G given byP = {x € G|0(x) = y(x)}.

Let H be a nonempty subset of a group (G,∗) is a subgroup of G iff for every a,b∈H imply ab−1∈H.

Answered: Determine whether the following subsets…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin square property to draw up the group table for G, and deduce that a3 = e
a in G such that y = a¯\xa. If x E Z(G), find [x], the equivalence class containing x. -1 ха. 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are in Z(G). 25. Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then
3. Let GL2 (R) be the group of all 2 x 2 nonsingular matrices with entries in R, and let SL2(Z) be the subset of GL2(R) given by SL2(Z) = {A € GL2(R) | all entries of A are in Z and det(A) = 1}. %3D (a) Prove that SL2(Z) is a subgroup of GL2(R). (b)) Let n be a positive integer, and let H be the subset of SL2(Z) given by 1+ np ng H : A E SL2(Z) | A: for some p, q, r, s E Z nr 1+ ns Prove that H is a normal subgroup of SL2(Z).
2.11 Let G be the set of all 2×2 matrices ), where a and b are nonzero real numbers. Show that G forms a group under matrix multiplication.
For any two elements x and y in a group G, there is a unique elementz in G such that y = xz.
Theorem 2. Let G; and G, be two groups. Let G = G,x G2 H = {(a,e,)|a e G} = G x{e;} %3D H, = {(e.b)beG} = {e}xG, and then G is an internal direct product of H and H,.
2.3. CONJUGACY CLASSES OF S AND An 33 Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4 Exercise 56. Prove that An is generated by permutations of type (12k), where 3 ≤ k ≤n. Exercise 57. Prove that the dihedral group De is isomorphic to the Sym- metric group S3. Exercise 58. Prove that Z(Sn) = 1. 2.3 Conjugacy classes of Sn and An By the results of the previous section, every permutation is a product of some cycles. The cyclic structure of a permutation shows the number of the cycles and their orders. For example, if o = (123) (45) (67,8,9), is a permutation in S12, then the cyclic structure of o is 132¹3¹4¹, which means o has 3 cycles of order 1, and on cycle of order2, one cycle of order 3
1. Let G be a group and let H, H, .., H, be the subgroups of G Then G is an internal direct product of H1, H, . H, if and only if the following conditions are satisfied : y.... H; 4G for i = 1, 2, .., n ..... (i) H;n IIH, ={e} jti (ii) G = H, H,.. H,
(a) Let G = (c, d| c4 = d² = (cd)² = e) and the subgroup H = (c|c4 = e) be the subgroup of group G. Find centralizer and normalizer of H in
Let (G, ◊) be a group and x = G. Suppose H is a subgroup of G that contains x. Which of the following must H also contain? Ox*, the inverse of x All elements x ◊ y for y = G OThe identity element e of G All "powers" x x, x0x 0x, ... -2 Enter the smallest subgroup of M₂(R)* containing the matrix (3²1) as a set. See formatting instructions below. To enter a matrix, type in its entries row by row top to bottom, left to right within each row (i.e. in "reading order") with single spaces separating the entries. You don't have to type the parentheses. So for a (2) you would type a b c d.
2. Let G be a finite group. Suppose H₁, H₂, . . . , Hk are subgroups of G that satisfy all of the following conditions: • H₂G for all i = 1,2,....k; |H₁|, |H₂|,. |H| are pairwise relatively prime; |H₁||H₂|· · · |Hk| = |G|. Show that G = H₁ H₂ ● H H₁ × H₂ × × Hk. k ≈

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