2.11. The group S3 consists of the following six distinct elements e, σ, σ', τ, στ, στ, where e is the identity element and multiplication is performed using the rules 0³ = = e, 7² = 1, TO = 0²T. Compute the following values in the group S3: (a) τσ (b) τ(στ) (c) (στ)(στ) Is S3 a commutative group? (d) (OT)(0²T). 2.12. Let G be a group, let d≥ 1 be an integer, and define a subset of G by G[d] = {g €G: gd = e}. (a) Prove that if g is in G[d], then g` is in G[d]. (b) Suppose that G is commutative. Prove that if 9₁ and 92 are in G[d], then their product 9₁ 92 is in G[d]. (c) Deduce that if G is commutative, then G[d] is a group. (d) Show by an example that if G is not a commutative group, then G[d] need not be a group. (Hint. Use Exercise 2.11.)
2.11. The group S3 consists of the following six distinct elements e, σ, σ', τ, στ, στ, where e is the identity element and multiplication is performed using the rules 0³ = = e, 7² = 1, TO = 0²T. Compute the following values in the group S3: (a) τσ (b) τ(στ) (c) (στ)(στ) Is S3 a commutative group? (d) (OT)(0²T). 2.12. Let G be a group, let d≥ 1 be an integer, and define a subset of G by G[d] = {g €G: gd = e}. (a) Prove that if g is in G[d], then g` is in G[d]. (b) Suppose that G is commutative. Prove that if 9₁ and 92 are in G[d], then their product 9₁ 92 is in G[d]. (c) Deduce that if G is commutative, then G[d] is a group. (d) Show by an example that if G is not a commutative group, then G[d] need not be a group. (Hint. Use Exercise 2.11.)
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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need help with 2.12
2.11 is a hint!
![### Group Theory Exercises
#### 2.11. The group \( S_3 \) consists of the following six distinct elements
\[ e, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau, \]
where \( e \) is the identity element and multiplication is performed using the rules
\[ \sigma^3 = e, \quad \tau^2 = 1, \quad \tau\sigma = \sigma^2\tau. \]
Compute the following values in the group \( S_3 \):
\[ (a) \quad \tau\sigma^2 \]
\[ (b) \quad \tau(\sigma\tau) \]
\[ (c) \quad (\sigma\tau)(\sigma\tau) \]
\[ (d) \quad (\sigma\tau)(\sigma^2\tau). \]
Is \( S_3 \) a commutative group?
#### 2.12. Let \( G \) be a group, let \( d \geq 1 \) be an integer, and define a subset of \( G \) by
\[ G[d] = \{ g \in G : g^d = e \}. \]
(a) Prove that if \( g \) is in \( G[d] \), then \( g^{-1} \) is in \( G[d] \).
(b) Suppose that \( G \) is commutative. Prove that if \( g_1 \) and \( g_2 \) are in \( G[d] \), then their product \( g_1 \star g_2 \) is in \( G[d] \).
(c) Deduce that if \( G \) is commutative, then \( G[d] \) is a group.
(d) Show by an example that if \( G \) is not a commutative group, then \( G[d] \) need not be a group. *(Hint: Use Exercise 2.11.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c5ad030-3ec8-4fd2-8d64-821b0d0d0877%2F10274ffd-dcf7-46a2-a0e0-228a37089888%2F6s7rxs9_processed.png&w=3840&q=75)
Transcribed Image Text:### Group Theory Exercises
#### 2.11. The group \( S_3 \) consists of the following six distinct elements
\[ e, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau, \]
where \( e \) is the identity element and multiplication is performed using the rules
\[ \sigma^3 = e, \quad \tau^2 = 1, \quad \tau\sigma = \sigma^2\tau. \]
Compute the following values in the group \( S_3 \):
\[ (a) \quad \tau\sigma^2 \]
\[ (b) \quad \tau(\sigma\tau) \]
\[ (c) \quad (\sigma\tau)(\sigma\tau) \]
\[ (d) \quad (\sigma\tau)(\sigma^2\tau). \]
Is \( S_3 \) a commutative group?
#### 2.12. Let \( G \) be a group, let \( d \geq 1 \) be an integer, and define a subset of \( G \) by
\[ G[d] = \{ g \in G : g^d = e \}. \]
(a) Prove that if \( g \) is in \( G[d] \), then \( g^{-1} \) is in \( G[d] \).
(b) Suppose that \( G \) is commutative. Prove that if \( g_1 \) and \( g_2 \) are in \( G[d] \), then their product \( g_1 \star g_2 \) is in \( G[d] \).
(c) Deduce that if \( G \) is commutative, then \( G[d] \) is a group.
(d) Show by an example that if \( G \) is not a commutative group, then \( G[d] \) need not be a group. *(Hint: Use Exercise 2.11.)*
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