Related questions
Question
Number 19
![filling in the details to establish the equality
12. Let G be a non-commutative group with 343 elements. Let Z be the center of G and
assume Z # {}. How many elements must Z contain? Explain your reasoning.
13. Let G be a group, and let Z be the center of G. Explain why [G: Z] is not a prime
number.
14. Prove: Let G be a finite group (so each g E G has finite order), and let H be a
subgroup of G. If G/H has an element of order n, then G has an element of order
n.
15. (a) Write the contrapositive of the implication in Theorem 23.7.
(b) Explain how G = D4 serves as an example of the contrapositive from part (a).
16. Prove: Let G be a commutative group, and let H be its subgroup. If every element
h E H is a square in H (i.e., h = k² for some k € H) and every element aH = G/H
is a square in G/H (i.e., aH =
(bH)2 for some bH = G/H), then every element of
G is a square in G.
17. Prove: Let G be a group, and let H be its subgroup. Suppose every element of H
and G/H has order 3" where n is a non-negative integer. Then every element of G
also has order 3" where n ≥ 0.
18. Let H be a subgroup of G with [G: H] = 2. Prove that aH = Ha for all a € G.
(This is the statement of Theorem 24.10.)
19. Let G be a group, and let H be a subgroup with [G: H] = 8. If g E G has odd order
(i.e., ord (g) is odd), then g € H.](https://content.bartleby.com/qna-images/question/47675d24-ba7f-4e21-93eb-97faaaed8f8d/31ac86b0-d0e8-40b0-ba39-d1e13d3dce09/08q4qre6_processed.jpeg)
Transcribed Image Text:filling in the details to establish the equality
12. Let G be a non-commutative group with 343 elements. Let Z be the center of G and
assume Z # {}. How many elements must Z contain? Explain your reasoning.
13. Let G be a group, and let Z be the center of G. Explain why [G: Z] is not a prime
number.
14. Prove: Let G be a finite group (so each g E G has finite order), and let H be a
subgroup of G. If G/H has an element of order n, then G has an element of order
n.
15. (a) Write the contrapositive of the implication in Theorem 23.7.
(b) Explain how G = D4 serves as an example of the contrapositive from part (a).
16. Prove: Let G be a commutative group, and let H be its subgroup. If every element
h E H is a square in H (i.e., h = k² for some k € H) and every element aH = G/H
is a square in G/H (i.e., aH =
(bH)2 for some bH = G/H), then every element of
G is a square in G.
17. Prove: Let G be a group, and let H be its subgroup. Suppose every element of H
and G/H has order 3" where n is a non-negative integer. Then every element of G
also has order 3" where n ≥ 0.
18. Let H be a subgroup of G with [G: H] = 2. Prove that aH = Ha for all a € G.
(This is the statement of Theorem 24.10.)
19. Let G be a group, and let H be a subgroup with [G: H] = 8. If g E G has odd order
(i.e., ord (g) is odd), then g € H.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 3 steps with 1 images
