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There is a unique group law on the 8- element set Q = {±1, ±i, ±j, ±k} that makes i2 = j2 = k2 = ijk = -1. Right and left multiplication by 1 and -1 are defined as usual. For example, (-1)i = -i = i(-1) and (-1)^2 = 1. (Q is called the quaternion group.) (a) Which of the 8 elements of Q is equal to i^-1? What about -jkj? %3D (b) Compute the orders of all the elements of Q. (c) Provide brief justification for why the subgroup H = {±1}is normal. (d) List a set of distinct coset representatives for Q/H (with H as above). %3D What is the order of each coset in the group Q/H? (e) Compare your answers from parts (b) and (d). Show that Q does not have a subgroup isomorphic to Q/H. Let Q and H be as in the previous question. (a) List all subgroups of Q/H. (b) What subgroups of Q correspond to the subgroups from part (a) under the Correspondence Theorem? (c) Show that every subgroup of Q is normal, even though Q is not abelian.

There is a unique group law on the 8- element set Q = {±1, ±i, ±j, ±k} that makes i2 = j2 = k2 = ijk = -1. Right and left multiplication by 1 and -1 are defined as usual. For example, (-1)i = -i = i(-1) and (-1)^2 = 1. (Q is called the quaternion group.) (a) Which of the 8 elements of Q is equal to i^-1? What about -jkj? %3D (b) Compute the orders of all the elements of Q. (c) Provide brief justification for why the subgroup H = {±1}is normal. (d) List a set of distinct coset representatives for Q/H (with H as above). %3D What is the order of each coset in the group Q/H? (e) Compare your answers from parts (b) and (d). Show that Q does not have a subgroup isomorphic to Q/H. Let Q and H be as in the previous question. (a) List all subgroups of Q/H. (b) What subgroups of Q correspond to the subgroups from part (a) under the Correspondence Theorem? (c) Show that every subgroup of Q is normal, even though Q is not abelian.

Advanced Engineering Mathematics
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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There is a unique group law on the 8- element set Q = {±1, ±i, ±j, ±k} that makes i2 = j2 = k2 = ijk = -1. Right and left multiplication by 1 and -1 are defined as usual. For example, (-1)i = -i = i(-1) and (-1)^2 = 1. (Q is called the quaternion group.) (a) Which of the 8 elements of Q is equal to i^-1? What about -jkj? %3D (b) Compute the orders of all the elements of Q. (c) Provide brief justification for why the subgroup H = {±1}is normal. (d) List a set of distinct coset representatives for Q/H (with H as above). What is the order of each coset in the group Q/H? (e) Compare your answers from parts (b) and (d). Show that Q does not have a subgroup isomorphic to Q/H. Let Q and H be as in the previous question. (a) List all subgroups of Q/H. (b) What subgroups of Q correspond to the subgroups from part (a) under the Correspondence Theorem? (c) Show that every subgroup of Q is normal, even though Q is not abelian.
Transcribed Image Text:There is a unique group law on the 8- element set Q = {±1, ±i, ±j, ±k} that makes i2 = j2 = k2 = ijk = -1. Right and left multiplication by 1 and -1 are defined as usual. For example, (-1)i = -i = i(-1) and (-1)^2 = 1. (Q is called the quaternion group.) (a) Which of the 8 elements of Q is equal to i^-1? What about -jkj? %3D (b) Compute the orders of all the elements of Q. (c) Provide brief justification for why the subgroup H = {±1}is normal. (d) List a set of distinct coset representatives for Q/H (with H as above). What is the order of each coset in the group Q/H? (e) Compare your answers from parts (b) and (d). Show that Q does not have a subgroup isomorphic to Q/H. Let Q and H be as in the previous question. (a) List all subgroups of Q/H. (b) What subgroups of Q correspond to the subgroups from part (a) under the Correspondence Theorem? (c) Show that every subgroup of Q is normal, even though Q is not abelian.
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