List of top Linear Algebra Questions on Group Theory asked in GATE MA

Consider the following conditions on two proper non-zero ideals \( J_1 \) and \( J_2 \) of a non-zero commutative ring \( R \): P: For any \( r_1, r_2 \in R \), there exists a unique \( r \in R \) such that \( r - r_1 \in J_1 \) and \( r - r_2 \in J_2 \). Q: \( J_1 + J_2 = R \) Then, which of the following statements is TRUE?

The number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^4$ is __________.

The number of subgroups of a cyclic group of order 12 is __________.

Let \( K \) denote the subset of \( \mathbb{C} \) consisting of elements algebraic over \( \mathbb{Q} \). Then, which of the following statements are TRUE?