Consider the group G = {1, −1, i, −i, j, −j, k, −k} under multiplication. Here...

Consider the group G = {1, −1, i, −i, j, −j, k, −k} under multiplication. Here
i2= j2= k2= ijk = −1. determine which of the following sets is a subgroup
of G. If a set is not a subgroup, give one reason why it is not.
(a) {1, −1}

(b) {i, −i, j, −j}

(c) {1, −1, i, −i}

(d) {1, i, −i, j}

Expert Solution

Colby Messinger answered 5 months ago

There exists a group G of order 8 having the following presentation: G=〈i,j,k | ij=k, jk=I,...

There exists a group G of order 8 having the following presentation: G=〈i,j,k | ij=k, jk=I, ki=j, i^2 =j^2 =k^2〉. Denotei2 bym. Showthat every element of G can be written in the form e, i, j, k, m, mi, mj, mk, and hence that these are precisely the distinct elements of G. Furthermore, write out the multiplication table for G (really, this should be going on while you do the first part of the problem).

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that...

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that (p − 1)2 = 1 (mod p) 2. Is the above true for any number (not necessarily prime)? 3. Show that the equation a 2 − 1 = 0, has only two solutions mod p. 4. Consider (p − 1)!. Show that (p − 1)! = −1 (mod p) Remark: Think about what are the values of inverses of 1, 2, . . ....

If p is prime, then Z*p is a group under multiplication.

Let G be an abelian group and K is a subset of G. if K is...

Let G be an abelian group and K is a subset of G. if K is a subgroup of G , show that G is finitely generated if and only if both K and G/K are finitely generated.

(1) Let G be a group and H, K be subgroups of G. (a) Show that...

(1) Let G be a group and H, K be subgroups of G. (a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a subgroup of G. (b) Show that if H and K are both normal subgroups, then HK is also a normal subgroup. (c) Give an example of subgroups H and K such that HK is not a subgroup of G.

import java.util.*; class A { int i, j, k; public A(int i, int j, int k)...

import java.util.*; class A { int i, j, k; public A(int i, int j, int k) { this.i=i; this.j=j; this.k=k; } public String toString() { return "A("+i+","+j+","+k+")"; } } class Main { public static void main(String[] args) { ArrayList<A> aL=new ArrayList<A>(); Random rand= new Random(1000); //1000 is a seed value for (int p=0; p<10; p++) { int i = rand.nextInt(100); int j = rand.nextInt(200); int k = rand.nextInt(300); aL.add(new A(i, j, k)); } System.out.println("----- Original arraylist------"); for (A a: aL)...

Consider the mathematical system with the single element {1} under the operation of multiplication. Answer and...

Consider the mathematical system with the single element {1} under the operation of multiplication. Answer and explain the following: a) Is the system closed? b) Does the system have an identity element? c) Does 1 have an inverse? d) Does the associative property hold? e) Does the commutative property hold? f) Is {1} under the operation of multiplication a commutative group?

air behaves as an ideal gas with R = 0.287 k J k g K. A...

air behaves as an ideal gas with R = 0.287 k J k g K. A compressor operates at steady state and takes in air from ambient 0 kPa, gage and 300 K. The outlet pressure is 60 kPa, gage and 300 K. Determine: the mass flow rate if the inlet area is 10 cm2 and the inlet pressure is -3 kPa, gage. the minimum outlet temperature that is possible for this compressor. the isentropic efficiency of the compressor, assuming...

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈...

a symmetric group S5 acts on the set X5 = {(i, j) : i, j ∈ {1, 2, 3, 4, 5}}. S5 will also act on this set. Consider the subgroup H = <(1, 2)(3, 4), (1, 3)(2, 4)>≤ S5. (a) Find the orbits of H in this action. Justify your answers. (b) For each orbit find the stabiliser one of its members. Justify your answers. action is this t.(i,j)=(t(i),t(j))

Prove that n, nth roots of unity form a group under multiplication.

Question