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(1) If the group G has 42 elements, the group H has 10 elements, : G H is a group homomorphism, and ker G, then what are the possibilities for the |ker()|? (m) Let G = As, the alternating group of order 60, and H = Z/120Z, the cyclic group of order 120. If o: G-H is a group homomorphism, then what are the possibilities for |ker()|? (n) Assume that the group G acts on a set with 5 elements and H is the set of elements of G that act trivially (i.e., h z=r for all hЄ H and r€ ). Then S, is guaranteed to have a subgroup isomorphic to which one(s) of the following: G, H, G/H?

(1) If the group G has 42 elements, the group H has 10 elements, : G H is a group homomorphism, and ker G, then what are the possibilities for the |ker()|? (m) Let G = As, the alternating group of order 60, and H = Z/120Z, the cyclic group of order 120. If o: G-H is a group homomorphism, then what are the possibilities for |ker()|? (n) Assume that the group G acts on a set with 5 elements and H is the set of elements of G that act trivially (i.e., h z=r for all hЄ H and r€ ). Then S, is guaranteed to have a subgroup isomorphic to which one(s) of the following: G, H, G/H?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 16E: For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have...
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(3) More Short answer questions. Name: (1) We know [G] = 80, the subgroup H G has 20 elements, and NG(H) has 40 elements. In the conju- gation action of G on the set of subgroups of G, what are the possibilities for the size of the orbit of H? (j) If G = Z/120Z and H = (10), then what is a familiar group isomorphic to G/H? (k) If H and K are subgroups of a group G, then for HK to also be a subgroup of G, is it necessary for one of H or K to be normal in G? (1) If the group G has 42 elements, the group H has 10 elements, : G H is a group homomorphism, and ker G, then what are the possibilities for the |ker()|? (m) Let G = A5, the alternating group of order 60, and H = Z/120Z, the cyclic group of order 120. If o: G→H is a group homomorphism, then what are the possibilities for |ker()|? (n) Assume that the group G acts on a set with 5 elements and H is the set of elements of G that act trivially (i.e., h x = x for all hЄ H and r€). Then S5 is guaranteed to have a subgroup isomorphic to which one(s) of the following: G, H, G/H? (o) If the group G has 35 elements, then what is the highest o(g) that we are guaranteed to have for g = G7 9 (p) If |G|= 112 = 24x7, then what are the possibilities for the number of Sylow 7-subgroups of G?
Transcribed Image Text:(3) More Short answer questions. Name: (1) We know [G] = 80, the subgroup H G has 20 elements, and NG(H) has 40 elements. In the conju- gation action of G on the set of subgroups of G, what are the possibilities for the size of the orbit of H? (j) If G = Z/120Z and H = (10), then what is a familiar group isomorphic to G/H? (k) If H and K are subgroups of a group G, then for HK to also be a subgroup of G, is it necessary for one of H or K to be normal in G? (1) If the group G has 42 elements, the group H has 10 elements, : G H is a group homomorphism, and ker G, then what are the possibilities for the |ker()|? (m) Let G = A5, the alternating group of order 60, and H = Z/120Z, the cyclic group of order 120. If o: G→H is a group homomorphism, then what are the possibilities for |ker()|? (n) Assume that the group G acts on a set with 5 elements and H is the set of elements of G that act trivially (i.e., h x = x for all hЄ H and r€). Then S5 is guaranteed to have a subgroup isomorphic to which one(s) of the following: G, H, G/H? (o) If the group G has 35 elements, then what is the highest o(g) that we are guaranteed to have for g = G7 9 (p) If |G|= 112 = 24x7, then what are the possibilities for the number of Sylow 7-subgroups of G?
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