**(1).** Let $ G $ be a group and $ a \in G $ be a certain fixed element of $ G $. The centralizer of $ a $ in $ G $ is \[C(a) = \{ g \in G \mid ga = ag \}\]i.e., it is a set of all elements in $ G $ that commute with $ a $. Then use either one-step or two-step subgroup test to show\[C(a) \leq G\]**(2).** $ \mathbb{Z}_{20} $ is a group under addition modulo 20. Provided that $ K \leq \mathbb{Z}_{20} $, $ 12, 16 \in K $ and moreover, $ K $ has the smallest order among all possible subgroups of $ \mathbb{Z}_{20} $ containing 12 and 16. Then list all elements of $ K $.**(3).** Let $ G = GL(2, \mathbb{R}) $. Then find the centralizer \[C \left( \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \right)\]

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Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g € G|ga = ag} i.e., it is a set of all element in G that commute with a. Then use either one-step or two-step subgroup test to show C(a) < G

Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g € G|ga = ag} i.e., it is a set of all element in G that commute with a. Then use either one-step or two-step subgroup test to show C(a) < G

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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Question

**(1).** Let $ G $ be a group and $ a \in G $ be a certain fixed element of $ G $. The centralizer of $ a $ in $ G $ is

\[
C(a) = \{ g \in G \mid ga = ag \}
\]

i.e., it is a set of all elements in $ G $ that commute with $ a $. Then use either one-step or two-step subgroup test to show

\[
C(a) \leq G
\]

**(2).** $ \mathbb{Z}_{20} $ is a group under addition modulo 20. Provided that $ K \leq \mathbb{Z}_{20} $, $ 12, 16 \in K $ and moreover, $ K $ has the smallest order among all possible subgroups of $ \mathbb{Z}_{20} $ containing 12 and 16. Then list all elements of $ K $.

**(3).** Let $ G = GL(2, \mathbb{R}) $. Then find the centralizer

\[
C \left( \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \right)
\]

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