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The question is: [ 7. Prove that the only nonzero characters of a group $ G $ which are homomorphisms are the linear characters.] Provide answer according this chegg guidelines: Solve this and ensure it is fully addressed: Create a detailed, step-by-step solution for a calculation-based question, regardless of the subject (e.g., math, physics, chemistry, etc..). The solution should be provided by an expert and adhere to the following guidelines: 1. Relevancy: The solution must directly address the question without extraneous information. 2. Completeness: Include every step, calculation, and explanation; do not skip any steps, even if they seem basic. 3. Accuracy: Ensure all calculations are correct, using the appropriate formulas and units, and provide a precise final answer. 4. Clarity: Use simple language, logically connect the steps, and make the solution easy to understand. 5. Structure: Begin with a restatement of the problem, followed by the steps, and conclude with the final answer. 6. Voice: Maintain a professional and educational tone, as if a teacher is explaining the solution. Provide the solution in English. Use LaTeX for all mathematical expressions (e.g., \int_0^1 x^2 dx = \frac{1}{3}). If the question has multiple parts, solve all of them. Show every step and mention any theorems or standard results used. The solution should be self-contained, requiring no external references. Please do not use personal pronouns such as I, We, and You, etc..... it must be human touch. Please highlight imp terms in during answering. Act like a normal, thoughtful person who’s knowledgeable but doesn’t write like a robot. I want your responses to feel conversational, relatable, and human. Not like a formal essay or a customer support script. Please avoid the following common mistakes: Don’t follow the same rigid structure in every reply (intro, bullets, summary). Don’t over-explain. Be concise when the answer is simple. Avoid using repetitive connectors like “however,” “on the other hand,” or “nevertheless” too much. Keep the tone balanced, not overly cheerful or overly formal. Don’t confidently state anything you’re unsure about. If something might be wrong, say so. Don’t flatter me unnecessarily. Keep it real. If you're speaking aloud (voice mode), don’t overuse unnatural filler sounds like “umm” or dramatic pauses. Try to use natural expressions and regional or cultural nuance when appropriate. Be sensitive to subtle context. Don’t miss the point or default to generic answers. Avoid em dashes. Please provide its Subject, Sub-Subject and topic.

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The question is: [ 7. Prove that the only nonzero characters of a group $ G $ which are homomorphisms are the linear characters.] Provide answer according this chegg guidelines: Solve this and ensure it is fully addressed: Create a detailed, step-by-step solution for a calculation-based question, regardless of the subject (e.g., math, physics, chemistry, etc..). The solution should be provided by an expert and adhere to the following guidelines: 1. Relevancy: The solution must directly address the question without extraneous information. 2. Completeness: Include every step, calculation, and explanation; do not skip any steps, even if they seem basic. 3. Accuracy: Ensure all calculations are correct, using the appropriate formulas and units, and provide a precise final answer. 4. Clarity: Use simple language, logically connect the steps, and make the solution easy to understand. 5. Structure: Begin with a restatement of the problem, followed by the steps, and conclude with the final answer. 6. Voice: Maintain a professional and educational tone, as if a teacher is explaining the solution. Provide the solution in English. Use LaTeX for all mathematical expressions (e.g., \int_0^1 x^2 dx = \frac{1}{3}). If the question has multiple parts, solve all of them. Show every step and mention any theorems or standard results used. The solution should be self-contained, requiring no external references. Please do not use personal pronouns such as I, We, and You, etc..... it must be human touch. Please highlight imp terms in during answering. Act like a normal, thoughtful person who’s knowledgeable but doesn’t write like a robot. I want your responses to feel conversational, relatable, and human. Not like a formal essay or a customer support script. Please avoid the following common mistakes: Don’t follow the same rigid structure in every reply (intro, bullets, summary). Don’t over-explain. Be concise when the answer is simple. Avoid using repetitive connectors like “however,” “on the other hand,” or “nevertheless” too much. Keep the tone balanced, not overly cheerful or overly formal. Don’t confidently state anything you’re unsure about. If something might be wrong, say so. Don’t flatter me unnecessarily. Keep it real. If you're speaking aloud (voice mode), don’t overuse unnatural filler sounds like “umm” or dramatic pauses. Try to use natural expressions and regional or cultural nuance when appropriate. Be sensitive to subtle context. Don’t miss the point or default to generic answers. Avoid em dashes. Please provide its Subject, Sub-Subject and topic.

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Asked by: Ruhail Khan
Created at: 2025-08-27 03:43:30

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**Subject:** Mathematics **Sub-Subject:** Abstract Algebra (Group Theory) **Topic:** Group Representations — Characters and Linear Characters --- **Restatement of the Problem:** Prove that the only nonzero characters of a group \( G \) which are homomorphisms are the linear characters. --- ### Step-by-Step Solution #### **Step 1: Understanding Characters and Linear Characters** - **Character**: In group representation theory, a *character* of a group \( G \) is a function \( \chi: G \to \mathbb{C} \) that arises as the trace of a group representation; that is, for a representation \( \rho: G \to GL(V) \), the character is \( \chi(g) = \text{Tr}(\rho(g)) \). - **Linear Character**: A *linear character* is a group homomorphism from \( G \) to the multiplicative group of nonzero complex numbers \( \mathbb{C}^\times \). That is, \( \lambda: G \to \mathbb{C}^\times \) such that \( \lambda(g_1 g_2) = \lambda(g_1)\lambda(g_2) \) for all \( g_1, g_2 \in G \). #### **Step 2: What Does It Mean for a Character to Be a Homomorphism?** Suppose \( \chi: G \to \mathbb{C} \) is a nonzero character that is also a group homomorphism, i.e., \( \chi(g_1g_2) = \chi(g_1)\chi(g_2) \) for all \( g_1, g_2 \in G \). #### **Step 3: Characters as Traces of Representations** A character \( \chi \) arises from some representation \( \rho: G \to GL(V) \), with \[ \chi(g) = \operatorname{Tr}(\rho(g)) \] The degree (or dimension) of \( \chi \) is \( n = \chi(1) = \dim V \). #### **Step 4: Homomorphism Condition on the Character** Given \( \chi \) is a homomorphism: \[ \chi(g_1g_2) = \chi(g_1)\chi(g_2) \] But also, \[ \chi(g_1g_2) = \operatorname{Tr}(\rho(g_1g_2)) = \operatorname{Tr}(\rho(g_1)\rho(g_2)) \] Generally, the trace of a product is not the product of the traces, except in trivial or special cases. #### **Step 5: When Is \(\operatorname{Tr}(AB) = \operatorname{Tr}(A)\operatorname{Tr}(B)\)?** Let \( n \) be the degree (the dimension of the representation). For arbitrary \( n \), trace is only multiplicative in one-dimensional representations (i.e., \( n = 1 \)). Otherwise, for higher dimensions, the trace is not multiplicative. #### **Step 6: Conclude Only Linear Characters Satisfy the Condition** So, if \( \chi \) is a nonzero homomorphism under multiplication, it must come from a one-dimensional representation. In other words, \( \rho: G \to GL(1, \mathbb{C}) = \mathbb{C}^\times \), and then \[ \chi(g) = \operatorname{Tr}(\rho(g)) = \rho(g) \] because the trace of a \( 1 \times 1 \) matrix is just the entry itself. Hence, \( \chi \) is a linear character. #### **Step 7: Any Nonzero Character Homomorphism Is a Linear Character** Summarizing, any nonzero character which is also a group homomorphism must be a linear character. Conversely, all linear characters are nonzero (since the image is in \( \mathbb{C}^\times \)) and are group homomorphisms. --- **Final Answer:** The only nonzero characters of a group \( G \) which are homomorphisms are the **linear characters**—that is, the group homomorphisms from \( G \) to \( \mathbb{C}^\times \). This follows because only one-dimensional representations (linear characters) have characters for which \( \chi(g_1g_2) = \chi(g_1)\chi(g_2) \), as for higher dimensions the trace function is not multiplicative.

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