(i) Let T be the subgroup of GL(2, ℂ) generated by [ ω 0 0 ω^2 ] and [ 0

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(i) Let T be the subgroup of GL(2, ℂ) generated by [ ω ...

(i) Let $T$ be the subgroup of $\mathrm{GL}(2, \mathbb{C})$ generated by $\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega^{2}\end{array}\right]$ and $\left[\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right]$, where $\omega=e^{2 \pi i / 3}$ is a primitive cube root of unity. Prove that $|T|=12$. (ii) Prove that $T$ has a presentation $$ \left(a, b \mid a^{6}=1, b^{2}=a^{3}=(a b)^{2}\right) $$ (iii) Prove that $T \cong \mathbb{I}_{3} \times \mathbb{I}_{4}$. Hint. Let $K=\langle u\rangle \cong \mathbb{I}_{3}$, let $Q=\langle x\rangle \cong \mathbb{I}_{4}$, and make $K$ into a $Q$-module by $x u=2 u, x(2 u)=u$, and $x^{2} u=u .$ In $K \times Q$, define $a=\left(2 u, x^{2}\right)$ and $b=(0, x) .$ (iv) Prove that every group $G$ of order 12 is isomorphic to exactly one of the following five groups: $$ \mathbb{I}_{12}, \quad \mathbf{V} \times \mathbb{I}_{3}, \quad A_{4}, \quad S_{3} \times \mathbb{I}_{2}, \quad T . $$

(i) Let $T$ be the subgroup of $\mathrm{GL}(2, \mathbb{C})$ generated by $\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega^{2}\end{array}\right]$ and $\left[\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right]$, where $\omega=e^{2 \pi i / 3}$ is a primitive cube root of unity. Prove that $|T|=12$.
(ii) Prove that $T$ has a presentation
$$
\left(a, b \mid a^{6}=1, b^{2}=a^{3}=(a b)^{2}\right)
$$
(iii) Prove that $T \cong \mathbb{I}_{3} \times \mathbb{I}_{4}$.
Hint. Let $K=\langle u\rangle \cong \mathbb{I}_{3}$, let $Q=\langle x\rangle \cong \mathbb{I}_{4}$, and make $K$ into a $Q$-module by $x u=2 u, x(2 u)=u$, and $x^{2} u=u .$ In $K \times Q$, define $a=\left(2 u, x^{2}\right)$ and $b=(0, x) .$
(iv) Prove that every group $G$ of order 12 is isomorphic to exactly one of the following five groups:
$$
\mathbb{I}_{12}, \quad \mathbf{V} \times \mathbb{I}_{3}, \quad A_{4}, \quad S_{3} \times \mathbb{I}_{2}, \quad T .
$$

Key Concepts

Subgroups of General Linear Groups

Subgroups of GL(2, ?), the group of invertible 2×2 complex matrices, are significant in the study of linear representations of groups. Such subgroups often serve as concrete examples of abstract group concepts and help illustrate properties like group order, element orders, and group actions in a setting that connects algebra with geometry. This concept is especially relevant when matrices represent group elements via rotations, reflections, or other symmetries, tying linear algebra and group theory together.

Finite Group Order and Counting Techniques

Understanding the order of a finite group is fundamental in group theory. This concept involves determining the number of elements in a group, often by analyzing the generators and relations or by using counting arguments derived from group actions and the structure of subgroups. In problems where a group is defined via specific generators, one typically uses the orders of these generators and their interactions to deduce the overall group order.

Group Presentations and Defining Relations

A group presentation is a way of describing a group by specifying a set of generators and a set of relations between those generators. This method encapsulates the structure of the group in a concise form, making it easier to work with abstract groups. By analyzing the given relations, such as specific powers of elements being equal to the identity or products of generators yielding other powers, one can compare the presentation with known groups or deduce properties like the group order.

Group Isomorphism and Classification

Group isomorphism is the idea that two groups are structurally the same if there exists a bijective homomorphism between them. Classification of finite groups, especially for a given order, involves determining all possible group structures (up to isomorphism) that a group of that order can have. This concept is crucial when the task is to show that a given group is isomorphic to a well-known group or to precisely determine its place in the classification of groups of a certain order.

Semidirect Product Structures

A semidirect product is a construction that generalizes the direct product of groups and is used to build new groups from known ones, especially when one subgroup acts on another. This concept is important when describing groups that result from nontrivial interactions between subgroups. In problems where a group is constructed from two subgroups with a specified action on one another, the semidirect product framework provides the necessary language and tools to analyze and decompose the group structure.

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