Quiz — Structure de groupe, IV.pdf

Question 1 sur 10 20:00

[{"id":34910,"question":"Quel est l'élément neutre d'un groupe (G, *) ?","option_a":"Un élément e tel que e * a = a * e = a pour tout a ∈ G","option_b":"Un élément e tel que e * a = a pour tout a ∈ G","option_c":"Un élément e tel que a * e = a pour tout a ∈ G","option_d":"Un élément e qui n'a pas d'inverse","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'élément neutre e d'un groupe vérifie e * a = a * e = a pour tout élément a du groupe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Un élément e tel que e * a = a * e = a pour tout a ∈ G\", \"b\":","_debug_options_count":4},{"id":34911,"question":"Un groupe est dit abélien si l'opération est commutative.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un groupe est abélien si pour tous éléments a et b, a * b = b * a. Cette propriété n'est pas toujours vérifiée dans un groupe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34912,"question":"Quel est l'ordre de l'élément (1 2 3) dans le groupe symétrique S₃ ?","option_a":"2","option_b":"3","option_c":"4","option_d":"6","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'ordre d'un élément est le plus petit entier k tel que l'élément élevé à la puissance k donne l'identité. Ici, (1 2 3)³ = id.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"6\"}}","_debug_options_count":4},{"id":34913,"question":"Un homomorphisme de groupes f : G → H est toujours injectif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un homomorphisme n'est pas nécessairement injectif. Seuls les isomorphismes (homomorphismes bijectifs) le sont.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34914,"question":"Quel est le groupe symétrique S₄ ?","option_a":"L'ensemble des permutations de 3 éléments","option_b":"L'ensemble des permutations de 4 éléments","option_c":"L'ensemble des rotations du carré","option_d":"L'ensemble des translations du plan","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le groupe symétrique Sₙ est l'ensemble des permutations de n éléments, muni de la composition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"L'ensemble des permutations de 3 éléments\", \"b\": \"L'ensemble de","_debug_options_count":4},{"id":34915,"question":"Un groupe cyclique est toujours fini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un groupe cyclique peut être fini (ex: ℤ\/nℤ) ou infini (ex: ℤ).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34916,"question":"Soit G un groupe et H un sous-groupe de G. Si a ∈ H et b ∈ H, alors a * b⁻¹ ∈ H.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette propriété caractérise un sous-groupe : H est stable par l'opération et contient les inverses de ses éléments.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34917,"question":"Quel est l'ordre du groupe symétrique S₃ ?","option_a":"3","option_b":"4","option_c":"6","option_d":"8","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'ordre de Sₙ est n! (factorielle de n). Pour n=3, 3! = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"3\", \"b\": \"4\", \"c\": \"6\", \"d\": \"8\"}}","_debug_options_count":4},{"id":34918,"question":"Un isomorphisme entre deux groupes préserve la structure mais pas nécessairement les éléments.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un isomorphisme est une bijection qui préserve la structure du groupe, mais les éléments eux-mêmes peuvent être différents.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":34919,"question":"Soit G un groupe fini d'ordre n. Quel est l'ordre d'un élément g ∈ G ?","option_a":"Un diviseur de n","option_b":"Un multiple de n","option_c":"Égal à n","option_d":"Toujours égal à 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par le théorème de Lagrange, l'ordre d'un élément divise l'ordre du groupe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Un diviseur de n\", \"b\": \"Un multiple de n\", \"c\": \"Égal à n\", \"d","_debug_options_count":4}]