Structures Algébriques - Groupe - Sous Groupe - Examen National 2025 SM Rattrapage

In this video, I'll go through Exercise 5 from Algebraic Structures with you. This exercise is taken from the 2025 national SM exam retake session, and covers internal composition laws, groups, subgroups, and group homomorphisms. This video is for second-year baccalaureate SM students. Join this channel to get access to exclusive benefits:    / @mathphys   Remember, it's important to try the exercise yourself before watching the solution. ▬▬▬▬▬▬▬ Donations ▬▬▬▬▬▬▬ ❤️ Support my channel: paypal.me/MathPhys77 ▬▬▬▬▬▬▬▬▬▬▬▬▬▬ 🔔 Please click the LIKE button if you enjoy our content, leave a comment, and subscribe to our YouTube channel to receive our new videos. 👉Write your questions in the comments section if you didn't understand something! 😀 💡Don't forget to SHARE with YOUR FRIENDS PLEASE!!💪 ▬▬▬▬▬▬▬▬▬▬▬Exercise▬▬▬▬▬ Full Exam: https://cutt.ly/otVJUvGP Exercise 4: Recall that (M3(R), +, x) is a noncommutative unital ring. We equip the set E = { x + yi / x∈Z and y∈Z } with the internal composition law * defined by: For all (x, y, x', y')∈Z^4 ; (x + yi) * (x' + y'i) = x + (-1)^y x' + (y + y')i Part I: a) Verify that: (1 - i) * (3 + 2i) = -2 + i b) Show that the operation * is not commutative in E. 2) Show that the operation * is associative in E. 3) Show that 0 is the identity element for the operation * in E. a) Verify that for all (x, y) belonging to Z^2; (x + yi) * ( (-1)^y ) * (-x - yi) = 0 b) Show that (E, *) is a noncommutative group. Part II: Let F = {x + 2yi / x∈Z and y∈Z} and G = {M(x, y) / x∈Z and y∈Z} a) Show that F is a subgroup of (E, *). b) Show that the operation * is commutative in F. a) Show that phi is a homomorphism from (F, *) to (M3(R), x). b) Show that: phi(F) = G. c) Deduce that (G, x) is a commutative group. 00:29 Exercise Statement 01:29 Question 1)a) 04:45 Question 1)b) 06:22 Question 2)a) 08:28 Question 2)b) 12:33 Question 3) 14:44 Question 4) 22:25 Question 5)a) 29:18 Question 5)b) 30:01 Question 6) ▬▬▬▬▬▬▬ SIMILAR VIDEOS ▬▬▬▬▬▬▬ Algebraic Structures - Groups - Subgroups - 2025 National Exam SM Retake:    • Structures Algébriques - Groupe - Sous Gro...   2025 National Exam SM - Algebraic Structures - Groups, Rings and Fields:    • Examen National 2025 SM - Structures Algéb...   Algebraic Structures - Integral Ring - SM Practice Exam - [Exercise 19]:    • Structures Algébriques - Anneau intègre - ...   Algebraic Structures - Group - Subgroup - 2024 SM National Exam Retake:    • Structures Algébriques - Groupe - Sous Gro...   2024 SM National Exam - Algebraic Structures - Group - Subgroup:    • Examen National 2024 SM - Structures Algéb...   Algebraic Structures - Group Morphism - 2023 SM National Exam Retake:    • Structures Algébriques - Morphisme de Grou...   2023 SM National Exam - Algebraic Structures - Groups, Ring, and Field - 2nd Year Baccalaureate SM:    • Examen National 2023 SM - Structures Algéb...   Algebraic Structures - Group Morphism - 2022 National Exam SM Retake:    • Structures Algébriques - Morphisme de Grou...   2022 National Exam SM - Algebraic Structures - Answer Key for the National Math Exam:    • Examen National 2022 SM - Structures Algéb...   Vector Spaces - Algebraic Structures - 2nd Year Bac SM - [Exercise 2]:    • Les Espaces Vectoriels - Structures Algébr...   Algebraic Structures - Group Morphism - 2nd Year Bac SM - [Lesson Part 6]:    • Les Structures Algébriques - Morphisme de ...   Algebraic Structures - Group - Subgroup - 2nd Year Bac SM - [Lesson Part 5]:    • Les Structures Algébriques - Groupe - Sous...   Structures Algebraic Structures - Group Morphism - SM Practice Exam - [Exercise 11]:    • Structures Algébriques - Morphisme de Grou...   Algebraic Structures - Groups, Rings, and Fields - 2nd Year Baccalaureate SM - [Exercise 10]:    • Structures Algébriques - Groupes Anneau et...   Algebraic Structures - Groups, Rings, and Fields - 2nd Year Baccalaureate SM - [Exercise 9]:    • Structures Algébriques - Groupes Anneau et...   Algebraic Structures - Group Morphism - 2020 National Exam SM:    • Structures Algébriques - Morphisme de Grou...   ALGEBRAIC STRUCTURES - Group Isomorphisms - Matrices - 2020 National Exam - 2nd Year Baccalaureate SM:    • STRUCTURES ALGÉBRIQUES - Isomorphisme de G...   ▬▬▬▬▬▬▬▬▬SOCIAL MEDIA▬▬▬▬▬▬▬▬▬▬ Subscribe here:    / @mathphys   Social Media Social media:   / 536279063660093     / math-phys-112225370317476     / math.phys   ▬▬▬▬▬▬▬▬▬▬Relevant subjects▬▬▬▬▬▬▬▬ high school diploma 2nd year high school diploma 2nd year high school diploma in mathematics and science high school diploma in mathematics and science high school diploma in biology and biology high school diploma in mathematics and science Mathematics and science final year of high school national exam ▬▬▬▬▬▬▬▬▬ Video summary▬▬▬▬▬▬▬▬▬ • Internal Composition Laws • Proving a noncommutative group. • Proving a subgroup • Group morphisms 👉 And you can check out the other videos in this playlist to see the other parts of this lesson and other exercises on "Algebraic Structures" and their applications. ▬▬▬▬▬▬▬▬ KEYWORDS ▬▬▬▬▬▬▬▬ #Algebraic_Structures #Group #National2025Retake