to consider two cases, when g has finite order and when g has infinite order) Le

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 Quanternion 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 determine 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.

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please ask them as individual questions

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