Explicitly give the elements and structure of the group Sn/An, n ≥5.
To solve for the elements and structure of the group \( S_n / A_n \) where \( n \geq 5 \), we are dealing with the quotient group formed by the symmetric group \( S_n \) and the alternating group \( A_n \), which is a normal subgroup of \( S_n \). ### Elements of \( S_n / A_n \): 1. Symmetric Group \( S_n \): The group of all permutations of \( n \) _________ ____ ___ ________ ___ ____ _______.
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जाँच कीजिए कि निम्नलिखित का एक उपसमूह है या नहीं:
i) ii)
पुष्टिकरण के साथ, किन्सी अक्रमविनिमेय समूह के एक क्रमाविनिमेय उपसमूह का उदाहरण दीजिए।
Consider the set X = R \{-1}. Define * on X by
X₁ * X2 = X + X2 + X₁X2 X1, X2 Ε Χ.
i) Check whether (X, *) is a group or not.
ii) Prove that x * x*x*...*x (n times) = (1+x)" -1∀ n∈ N and x ∈ X.
Prove that as rings.
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)
i) (n) = n-1∀n ∈ N, where o is the Euler-phi function.
ii) If G₁ and G₂ are groups, and f: G₁→ G₂ is a group homomorphism, then 2 o(G₁) = 0(G2).
If G is an abelian group, then G is cyclic.
iv) If G is a group and HAG, then | G: H=2.
v) Every element of S has order at most n.
vi) If R is a ring and I is an ideal of R, then xr = rx ∀ x ∈ I and re R.
vii) If σε Sn(n ≥3) is a product of an even number of disjoint cycles, then sign (σ)=1.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to R form a ring with respect to point- wise addition and multiplication.
फलन पर विचार कीजिए। मान लीजिए कि
तब फलन के संजन के सापेक्षा B एक समूह है। जाँच कीजिए कि
उपसमूह है या नहीं।
द्वारा
पर एक संबंध R परिभाषित कीजिए। जाँच कीजिए कि R एक तुल्यता संबंध है या नहीं। यदि है, तो सभी अलग-अलग तुल्यता वर्ग ज्ञात कीजिए। यदि R. एक तुल्यता संबंध नहीं है, तो
पर एक तुल्यता संबंध परिभाषित कीजिए।