Question

Question: Show that D₄ is not isomorphic to the quaternion group of Exercise 16 of Section 7.1.

Short answer

Answer

Itis proved that the groups D₄and Q₈are not isomorphic.

Step-by-step solution

Isomorphic Groups

Two groups G and Hare said to be isomorphic if there exist an isomorphism F from G onto H .

Find the order of elements of the group D4 

Consider the group D₄ .

D₄ = {R_{0 ,}R₉₀,R_180,R₂₇₀ H,V,D,D^I }

Find the order of elements of group D₄ .

Since, R₀^I = R₀ . Therefore, the order of is .

Find the order of R₉₀, .

R₉₀ ⁴=R₃₆₀

=R₀

Therefore, the order of R₉₀ is 4

Find the order of R_180, .

R₁₈₀ ²=R₃₆₀

=R₀

Therefore, the order of R₁₈₀ is 2.

Find the order of .R₂₇₀

R₂₇₀ ²=R₁₈₀

=R₀

But, it is known that R₁₈₀ ²=R₀. Then,

(R₂₇₀ )⁴ = R₀

Therefore, the order ofR₂₇₀ is 4 .

Every reflection is of order 2

H² = R₀ .

V² = R₀ .

D² = R₀ .

D^I2 = R₀ .

Therefore, the group D₄ has five elements {R_{0 ,}R₉₀,R_180,R₂₇₀ H,V,D,D^I } of order , R₉₀ andR₂₇₀ two elements of order 2 .

Find the order of elements of group Q8

Find the order of elements of group Q₈.

1 = 1

( - 1 )² = 1

( i )⁴ = 1

( - i)⁴ = 1

And,

( j )⁴ = 1

( - j)⁴ = 1

( k)⁴ = 1

( - k)⁴ = 1

Therefore, groupQ₈ has six elements$(\pm i,\pm j,\pm k)$ of order4 , and one element (-1) of order 2 .