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(7) The dihedral group is the subgroup D4 of S4 of order 8 consisting of permutations induced by symmetries of a square; in other words, it is generated by the elements (1234) (corresponding to a rotation of order 4) and (12)(34) (corresponding to a reflection). Show that D4 is not a normal subgroup of S4. (8) Show that D4 contains a subgroup of order 2 which is normal, and also a subgroup of order 2 which is not normal. (Hint: what is the center of D4?)

(7) The dihedral group is the subgroup D4 of S4 of order 8 consisting of permutations induced by symmetries of a square; in other words, it is generated by the elements (1234) (corresponding to a rotation of order 4) and (12)(34) (corresponding to a reflection). Show that D4 is not a normal subgroup of S4. (8) Show that D4 contains a subgroup of order 2 which is normal, and also a subgroup of order 2 which is not normal. (Hint: what is the center of D4?)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Abstract Algebra I 

**Problem 7:** The dihedral group is the subgroup \( D_4 \) of \( S_4 \) of order 8 consisting of permutations induced by symmetries of a square; in other words, it is generated by the elements \( (1234) \) (corresponding to a rotation of order 4) and \( (12)(34) \) (corresponding to a reflection). Show that \( D_4 \) is not a normal subgroup of \( S_4 \). **Problem 8:** Show that \( D_4 \) contains a subgroup of order 2 which *is* normal, and also a subgroup of order 2 which *is not* normal. (Hint: what is the center of \( D_4 \)?)
Transcribed Image Text:**Problem 7:** The dihedral group is the subgroup \( D_4 \) of \( S_4 \) of order 8 consisting of permutations induced by symmetries of a square; in other words, it is generated by the elements \( (1234) \) (corresponding to a rotation of order 4) and \( (12)(34) \) (corresponding to a reflection). Show that \( D_4 \) is not a normal subgroup of \( S_4 \). **Problem 8:** Show that \( D_4 \) contains a subgroup of order 2 which *is* normal, and also a subgroup of order 2 which *is not* normal. (Hint: what is the center of \( D_4 \)?)
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