Could you explain how to show 7.2 in detail? I also included lists of definitions and theorems in the book as a reference.Question : 7.2 Which permutations lie in the centralizer of (1 2 3) in S5? **7.2.** Which permutations lie in the centralizer of $(1\ 2\ 3)$ in $S_5$?**7.8.** If $a \in G$ and $a$ has odd order, show that $C(a) = C(a^4)$. **Centralizers and Normalizers in Group Theory****Definition 7.1:** Let $ G $ be a group, $ a \in G $, and $ H $ a subgroup of $ G $. The centralizer of $ a $ is the set of all elements of $ G $ that commute with $ a $. We denote this by $ C(a) = \{ g \in G : ag = ga \} $. Similarly, the centralizer of $ H $ is $ C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \} $.**Example 7.1:** If $ a \in Z(G) $, then $ C(a) = G $. If $ H \leq Z(G) $, then $ C(H) = G $. Notably, $ C(e) = G $, illustrating that centralizers aren't necessarily abelian.**Example 7.2:** Let $ G = D_8 $. Here, $ C(R_{270}) = \langle R_{90} \rangle $, $ C(R_{180}) = G $, and $ C(F_1) = \{ R_0, R_{180}, F_1, F_2 \} $.**Theorem 7.1:** Let $ G $ be a group, $ a \in G $, and $ H $ a subgroup.1. $ C(H) = \bigcap_{h \in H} C(h) $.2. $ C(a) $ and $ C(H) $ are both subgroups of $ G $.3. If $ H $ is a normal subgroup of $ G $, then so is $ C(H) $.4. $ Z(G) $ is a subgroup of both $ C(a) $ and $ C(H) $.5. $ C(a) = C(\langle a \rangle) $.**Definition 7.2:** Let $ G $ be a group and $ H $ a subgroup. The normalizer of $ H $ is the set $ N(H) = \{ a \in G : a^{-1}Ha = H \} $. If $ K $ is another subgroup of $ G $, then \( N_K(H) = N

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Description: Could you explain how to show 7.2 in detail? I also included lists of definitions and theorems in the book as a reference.Question : 7.2 Which permutations lie in the centralizer of (1 2 3) in S5? **7.2.** Which permutations lie in the centralizer of

we have to find the centralizer of element 1 2 3 in S5. Definition of Centralizer is, CS5=a∈S5ab=ba…

Answered: 7.2. Which permutations lie in the…

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7.2. Which permutations lie in the centralizer of (1 2 3) in S5?

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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Could you explain how to show 7.2 in detail? I also included lists of definitions and theorems in the book as a reference.

Question : 7.2 Which permutations lie in the centralizer of (1 2 3) in S5?

**7.2.** Which permutations lie in the centralizer of $(1\ 2\ 3)$ in $S_5$?

**7.8.** If $a \in G$ and $a$ has odd order, show that $C(a) = C(a^4)$.

**Centralizers and Normalizers in Group Theory**

**Definition 7.1:** Let $ G $ be a group, $ a \in G $, and $ H $ a subgroup of $ G $. The centralizer of $ a $ is the set of all elements of $ G $ that commute with $ a $. We denote this by $ C(a) = \{ g \in G : ag = ga \} $. Similarly, the centralizer of $ H $ is $ C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \} $.

**Example 7.1:** If $ a \in Z(G) $, then $ C(a) = G $. If $ H \leq Z(G) $, then $ C(H) = G $. Notably, $ C(e) = G $, illustrating that centralizers aren't necessarily abelian.

**Example 7.2:** Let $ G = D_8 $. Here, $ C(R_{270}) = \langle R_{90} \rangle $, $ C(R_{180}) = G $, and $ C(F_1) = \{ R_0, R_{180}, F_1, F_2 \} $.

**Theorem 7.1:** Let $ G $ be a group, $ a \in G $, and $ H $ a subgroup.

1. $ C(H) = \bigcap_{h \in H} C(h) $.
2. $ C(a) $ and $ C(H) $ are both subgroups of $ G $.
3. If $ H $ is a normal subgroup of $ G $, then so is $ C(H) $.
4. $ Z(G) $ is a subgroup of both $ C(a) $ and $ C(H) $.
5. $ C(a) = C(\langle a \rangle) $.

**Definition 7.2:** Let $ G $ be a group and $ H $ a subgroup. The normalizer of $ H $ is the set $ N(H) = \{ a \in G : a^{-1}Ha = H \} $. If $ K $ is another subgroup of $ G $, then \( N_K(H) = N

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