For each of the following groups G determine ez=the number of elements of order 5 and s5 = the number of subgroups of order 5. Give an example of each. G n5 | an order 5 element S5 an order 5 subgroup Z5 O Z5 U (25) Aut(Z25)
For each of the following groups G determine ez=the number of elements of order 5 and s5 = the number of subgroups of order 5. Give an example of each. G n5 | an order 5 element S5 an order 5 subgroup Z5 O Z5 U (25) Aut(Z25)
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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Abstract Algebra
![For each of the following groups \( G \), determine \( e_5 \) = the number of elements of order 5 and \( s_5 \) = the number of subgroups of order 5. Give an example of each.
\[
\begin{array}{|c|c|c|c|c|}
\hline
G & n_5 & \text{an order 5 element} & s_5 & \text{an order 5 subgroup} \\
\hline
\mathbb{Z}_5 \oplus \mathbb{Z}_5 & & & & \\
\hline
U(25) & & & & \\
\hline
\text{Aut}(\mathbb{Z}_{25}) & & & & \\
\hline
\end{array}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecfe7738-95ba-450b-a609-248aa7606e96%2F7b1adf9f-7774-4ac2-90f9-637a40c81fda%2F2bitnia_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each of the following groups \( G \), determine \( e_5 \) = the number of elements of order 5 and \( s_5 \) = the number of subgroups of order 5. Give an example of each.
\[
\begin{array}{|c|c|c|c|c|}
\hline
G & n_5 & \text{an order 5 element} & s_5 & \text{an order 5 subgroup} \\
\hline
\mathbb{Z}_5 \oplus \mathbb{Z}_5 & & & & \\
\hline
U(25) & & & & \\
\hline
\text{Aut}(\mathbb{Z}_{25}) & & & & \\
\hline
\end{array}
\]
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