(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?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just L-n please!
![(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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fbaae5-8d47-4476-8095-8b380294ae7e%2Faa9cf25c-2a26-4e68-b210-6f477cb6c687%2Fgt9ktxj_processed.jpeg&w=3840&q=75)
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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