SECTION 4.2 Basics of Group Theory 223
(i) Show that the six elementse, σ, σ^2 , τ, στ, σ^2 τ comprise all of
the elements of this group.
(ii) Show thatσ^3 =τ^2 =eand thatτσ=σ^2 τ.
(iii) From the above, complete the multiplication table:◦ e σ σ^2 τ στ σ^2 τ
e e σ σ^2 τ στ σ^2 τ
σ σ σ^2
σ^2 σ^2 τ
τ τ σ
στ στ e
σ^2 τ σ^2 τ- LetGbe the set of all 2×2 matrices with coefficients inZ 2 with de-
terminant 6 = 0. Assuming that multiplication is associative, show
thatGis a group of order 6. Next, set
A=
0 1
1 1
, B=
1 0
1 1
.Show that A^3 = B^2 =
1 0
0 1
(the identity of G), and thatBAB=A−^1.- Let (G,∗) be a group such that for every g∈ G,g^2 =e. Prove
thatGmust be an Abelian group. - LetZ 3 be the integers modulo 3 and consider the setUof matrices
with entries inZ 3 , defined by setting
U =
1 a c
0 1 b
0 0 1
∣∣
∣∣
∣∣
∣a, b, c∈Z 3
.
(a) Show thatU is a group relative to ordinary matrix multipli-
cation.
(b) Show that|U|= 27.