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We consider the group Ss. This is a group with 8! elements. Give an example of a 7-Sylow subgroup. (a) (b) How many 7-Sylow subgroups exist in Sg? Explain your answer. (I am not asking you to apply Sylow's Theorem here that occurs in (c). Here, you will be thinking very specifically about the group Ss and the example you created in (a). 2. (c) Sylow's Theorem gives partial information about the possible number of 7-Sylow in a group with 8! elements. Now, check that your answer in (b) is compatible with the information that comes from Sylow's Theorem.

We consider the group Ss. This is a group with 8! elements. Give an example of a 7-Sylow subgroup. (a) (b) How many 7-Sylow subgroups exist in Sg? Explain your answer. (I am not asking you to apply Sylow's Theorem here that occurs in (c). Here, you will be thinking very specifically about the group Ss and the example you created in (a). 2. (c) Sylow's Theorem gives partial information about the possible number of 7-Sylow in a group with 8! elements. Now, check that your answer in (b) is compatible with the information that comes from Sylow's Theorem.

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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We consider the group Sg. This is a group with 8! elements. Give an example of a 7-Sylow subgroup. (a) (b) How many 7-Sylow subgroups exist in Sg? Explain your answer. (I am not asking you to apply Sylow's Theorem here that occurs in (c). Here, you will be thinking very specifically about the group Ss and the example you created in (a). 2. (c) Sylow's Theorem gives partial information about the possible number of 7-Sylow in a group with 8! elements. Now, check that your answer in (b) is compatible with the information that comes from Sylow's Theorem. Now we move on to 2-Sylow subgroups: Since 27 128 is the largest power of 2 that divides 8!, a 2-Sylow subgroup of Sg will have 128 elements. Here is an example of such a subgroup: draw two congruent squares with the vertices labeled, such as 4 ما 8 Let H be the symmetries of this diagram. It includes two types of elements: . There are symmetries that preserve each of the two squares. For example, we could rotate the first square by 90 degrees clockwise, and reflect the second square over a horizontal line through the square. That would give (1234) (58) (67). You should realize that in this way, you get D4 x D4, which has 64 elements. . But there are another 64 elements that involve swapping the two squares. So for example, we could simply interchange the two squares (without rotating or reflecting either), giving the permutation (15) (26) (37) (48). As a more complicated example, we could rotate the first square by 180 degrees and drop it on the original second square, and reflect the second square over the diagonal between 6 and 8 and drop it onto the original first square. That would give (17) (2846) (35) (try it!). My questions for you: (d) How many 2-Sylow subgroups of Ss exist? Explain your answer. (e) Check that your answer to (d) is compatible with the information from Sylow's Theorem about the number of 2-Sylow subgroups in a group with 8! elements. (f) Use Sylow's Theorem to show why every 8-cycle is contained in some 2-Sylow subgroup, and why every 2-Sylow subgroup of S8 has to contain an 8-cycle. (g) Find an 8-cycle in the 2-Sylow subgroup I described for you! Explain your answer.
Transcribed Image Text:We consider the group Sg. This is a group with 8! elements. Give an example of a 7-Sylow subgroup. (a) (b) How many 7-Sylow subgroups exist in Sg? Explain your answer. (I am not asking you to apply Sylow's Theorem here that occurs in (c). Here, you will be thinking very specifically about the group Ss and the example you created in (a). 2. (c) Sylow's Theorem gives partial information about the possible number of 7-Sylow in a group with 8! elements. Now, check that your answer in (b) is compatible with the information that comes from Sylow's Theorem. Now we move on to 2-Sylow subgroups: Since 27 128 is the largest power of 2 that divides 8!, a 2-Sylow subgroup of Sg will have 128 elements. Here is an example of such a subgroup: draw two congruent squares with the vertices labeled, such as 4 ما 8 Let H be the symmetries of this diagram. It includes two types of elements: . There are symmetries that preserve each of the two squares. For example, we could rotate the first square by 90 degrees clockwise, and reflect the second square over a horizontal line through the square. That would give (1234) (58) (67). You should realize that in this way, you get D4 x D4, which has 64 elements. . But there are another 64 elements that involve swapping the two squares. So for example, we could simply interchange the two squares (without rotating or reflecting either), giving the permutation (15) (26) (37) (48). As a more complicated example, we could rotate the first square by 180 degrees and drop it on the original second square, and reflect the second square over the diagonal between 6 and 8 and drop it onto the original first square. That would give (17) (2846) (35) (try it!). My questions for you: (d) How many 2-Sylow subgroups of Ss exist? Explain your answer. (e) Check that your answer to (d) is compatible with the information from Sylow's Theorem about the number of 2-Sylow subgroups in a group with 8! elements. (f) Use Sylow's Theorem to show why every 8-cycle is contained in some 2-Sylow subgroup, and why every 2-Sylow subgroup of S8 has to contain an 8-cycle. (g) Find an 8-cycle in the 2-Sylow subgroup I described for you! Explain your answer.
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