to consider two cases, when g has finite order and when g has infinite order) Le
ID: 2973761 • Letter: T
Question
to consider two cases, when g has finite order and when g has infinite order) Let The set {plusminus1, plusminusI, plusminusJ, plusminus K) with operation matrix multiplication is called the Quaternion Group. List the subgroups of the Quaternion Group. Show that the set G = {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity element of this group? Is G a subgroup of a group (other than itself)? If so, what group? If not, why not? Construct the Cayley table for U(8). Can you see any relationship between U(8) and G? Prove that the set of all 2times2 matrices with entries from R and determinant 1 is a group under matrix multiplication by considering it as a subgroup of GL(2, R). This group is known as the "Special Linear Group" and is denoted SL(2, R). Let G be a group. Show that Z(G)={x G | gx=xg for g G} Is a subgroup of G. This group is called the center of G.Explanation / Answer
Quaternian group means
Q = {-1,i,j,k such that (-1)^2 = 1 i^2 = j^2 = k^2 = ijk = -1
where 1 is the identity element and -1 commutes with the other elements of the group.
in our case the identity element is identity matrix with rows and columns 2
check for different combinations of sets the values of i^2 j^2 ij etc and see the condition satisfies or not ...if it satisfies then its an sub group
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