Determine Group structure of given 2×2 matrix group over F3

Question

Determine  Group structure of given 2×2 matrix group over F3

Buck Henry · Accepted Answer

Step 1  I think that concentrating on the case p=3 and special properties of groups of order 12 isnt

reinosodairyshm · Accepted Answer

Step 1
I want to determined group structure of

G = {(\begin{array}{cc} a & b \\ 0 & d \end{array}) ∣ a, b \in F_{3}^{\times}, c \in F_{3}}

Step 2
For finite non-abelian groups of matrices over finite fields one tries to realize the group as a semi direct product of smaller abelian groups:
Let

H = {(\begin{array}{cc} 1 & c \\ 0 & 1 \end{array}) ∣ c \in F_{3}} \tilde{=} F_{3}

and

N = {(\begin{array}{cc} a & 0 \\ 0 & b \end{array}) ∣ a, b \in F_{3}^{⋆}} \tilde{=} {(F_{3}^{⋆})}^{2} \tilde{=} {(\frac{Z}{(2)})}^{2}

It follows for any

g \in G, h \in H

that

g h g^{- 1} \in H

hence

H \subseteq G

is a normal subgroup and the action σ of G on H is via the character

ρ : G \to F_{3}^{⋆}

defined by

ρ (g) = \frac{a}{b}

You get an action

σ : G \times H \to H

defined by

σ (g, x) = ρ (g) x = \frac{a}{b} x

with

x \in F_{3} and a, b \in F_{3}^{⋆}

Since

N H = G

and

N \cap H = {e}

you may express G as a semi direct product

G \tilde{=} N ⋊ H \tilde{=} {(F_{3}^{⋆})}^{2} ⋊_{ρ (-)} F_{3} \tilde{=} {(\frac{Z}{(2)})}^{2} ⋊_{ρ (-)} \frac{Z}{(3)}

This express G as a semi direct product of two known abelian groups - maybe this is helpful classifying the group. Im

alenahelenash · Accepted Answer

You have already determined that G is isomorphic to one of D12, Q12 or A4. You have also determined that G has two elements of order 6. As A4 has no elements of order 6, this option is eliminated. The group Q12 has an element of order 4, but G has no elements of order 4. This eliminates Q12, and so G≅D12.

Determine Group structure of given 2\times2 matrix group over \mathbb{F}_{3}

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