Question

2. The quaternion group is the group G = {1, -1, i, -i, j, -j, k, -k} with identity element 1 and defined by the following rules: (-1)^2 = 1, i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j, (-1)x = -x = x(-1) for all x ? G. (a) Show that the subgroup H = {1, -1} is normal in G. (b) Show that G/H ? Z2 ? Z2.

          2. The quaternion group is the group G = {1, -1, i, -i, j, -j, k, -k} with identity element 1 and defined by the following rules:
(-1)^2 = 1,
i^2 = j^2 = k^2 = -1,
ij = k, jk = i, ki = j,
ji = -k, kj = -i, ik = -j,
(-1)x = -x = x(-1) for all x ? G.
(a) Show that the subgroup H = {1, -1} is normal in G.
(b) Show that G/H ? Z2 ? Z2.

2. The quaternion group is the group G = 1, -1, i, -i, j, -j, k, -k with identity element 1 and defined by the following rules:
(-1)^2 = 1,
i^2 = j^2 = k^2 = -1,
ij = k, jk = i, ki = j,
ji = -k, kj = -i, ik = -j,
(-1)x = -x = x(-1) for all x ? G.
(a) Show that the subgroup H = 1, -1 is normal in G.
(b) Show that G/H ? Z2 ? Z2.

Added by Mandy H.

Mathematical Methods for Physics and Engineering

K. F. Riley, M. P. Hobson, S. J. Bence 3rd Edition

Chapter 28

Instant Answer

Solved by Expert Sri K

12/10/2022

Step 1

Step 1: To show that the subgroup H = {1, -1} is normal in G, we need to show that for any element g in G and any element h in H, the conjugate ghg^-1 is also in H. Show more…

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The quaternion group is the group G = {1, -1, i, -i, j, -j, k, -k} with identity element 1 and defined by the following rules: (-1)^2 = 1, i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j, (-1)x = -x = x(-1) for all x ∈ G. (a) Show that the subgroup H = {1, -1} is normal in G. (b) Show that G/H ≈ Z2 ⊕ Z2.