Order of GL₂(ℤ₉) Group

Let G be the group of 2 × 2 invertible matrices with entries from ℤ₉ under matrix multiplication. This group is also known as the General Linear Group of degree 2 over the ring ℤ₉, denoted as GL₂(ℤ₉). We are asked to find the order of this group, denoted by 'a', and determine which of the given statements about its divisibility are true.

The ring ℤ₉ is a commutative ring with unity. Since $9 = 3^2$, ℤ₉ is of the form ℤ_p^k, where $p=3$ and $k=2$. The size of the matrices is 2 × 2, so $n=2$.

Calculating the Order of GL₂(ℤ₉)

The formula for the order of the General Linear Group GL_n(ℤ_p^k) is given by:

$$|GL_n(\mathbb{Z}_{p^k})| = p^{k n^2 - n} \prod_{i=0}^{n-1} (p^n - p^i)$$

We have $p=3$, $k=2$, and $n=2$. Substituting these values into the formula:

$$a = |GL_2(\mathbb{Z}_{3^2})| = 3^{2 \cdot 2^2 - 2} \prod_{i=0}^{2-1} (3^2 - 3^i)$$ $$a = 3^{2 \cdot 4 - 2} (3^2 - 3^0)(3^2 - 3^1)$$ $$a = 3^{8 - 2} (9 - 1)(9 - 3)$$ $$a = 3^6 \cdot 8 \cdot 6$$

Next, we find the prime factorization of 8 and 6:

$$8 = 2^3$$ $$6 = 2 \cdot 3$$

Substitute these prime factors back into the expression for 'a':

$$a = 3^6 \cdot (2^3) \cdot (2^1 \cdot 3^1)$$ $$a = 3^6 \cdot 2^3 \cdot 2^1 \cdot 3^1$$

Combine the terms with the same base by adding their exponents:

$$a = 3^{6+1} \cdot 2^{3+1}$$ $$a = 3^7 \cdot 2^4$$

So, the order of the group G is $a = 3^7 \cdot 2^4$.

Analyzing Divisibility Statements

We need to check the divisibility statements based on the calculated order $a = 3^7 \cdot 2^4$.

• Statement 1: a is divisible by $3^4$.
  The prime factorization of $a$ is $3^7 \cdot 2^4$. For $a$ to be divisible by $3^4$, the exponent of 3 in the prime factorization of $a$ must be greater than or equal to 4. The exponent of 3 in $a$ is 7, and $7 \geq 4$. Thus, $a$ is divisible by $3^4$. This statement is true.
• Statement 2: a is divisible by $2^4$.
  The prime factorization of $a$ is $3^7 \cdot 2^4$. For $a$ to be divisible by $2^4$, the exponent of 2 in the prime factorization of $a$ must be greater than or equal to 4. The exponent of 2 in $a$ is 4, and $4 \geq 4$. Thus, $a$ is divisible by $2^4$. This statement is true.

The statements that are true are that $a$ is divisible by $3^4$ and $a$ is divisible by $2^4$.