O‘rin almashtirishlar uchun

2. Quyidagi α va β o‘rin almashtirishlar uchun α ◦ β ◦ α⁻¹ ifodani toping: • α = (1 2 5 7), β = (2 4 6) ∈ S₇.
• α = (1 3 5 7), β = (2 4 8) ◦ (1 3 6) ∈ S₈.
• α = (1 3) ◦ (5 8), β = (2 3 6 7) ∈ S₈.
• α = (2 5 9) ◦ (1 3 6), β = (1 5 7) ◦ (2 4 6 9) ∈ S₉.
3. (1 3 5 7) va (2 3 6 8) ∈ S₈ sikllar uchun α ◦ (1 3 5 7) ◦ α⁻¹ = (2 3 6 8) tenglikni qanoatlantiruvchi α o‘rin almashtirishni toping.
4. Quyidagi elementlarning tartiblarini aniqlang: • (1 2 3) ◦ (4 5) ∈ S₅.
• (1 2 4 3) ◦ (5 6) ∈ S₆.
• (1 7 4 3) ◦ (2 6 5) ∈ S₇. • (1 2 4 3) ◦ (2 6 5) ∈ S₆. • (1 2 7) ◦ (1 3 5) ∈ S₇.
5. Agar σ ∈ S_n o‘rin almashtirish o‘zaro kesishmaydigan sikllar ko‘paymasi ko‘rinishida
σ = σ₁ ◦ σ₂ ◦ · · · ◦ σ_k
kabi ifodalangan bo‘lib, ord(σ_i) = n_i, i ∈ {1, 2, . . . , k} bo‘lsa, u holda
ord(σ) = EKUB(n₁, n₂, . . . , n_k)
ekanligini isbotlang.
6. (1 2 . . . n − 1 n)⁻¹ = (n n − 1 . . . 2 1) tenglikni isbotlang.
7. α = (a₁ a₂ . . . a_k) ∈ S_n sikl berilgan bo‘lsin. U holda quyidagi tenglikni isbotlang:
. S₄ gruppaning tartibi ikkiga teng bo‘lgan barcha elementlarini toping. 9. S₄ gruppaning tartibi uchga teng bo‘lgan barcha elementlarini toping.
10. A₄ gruppaning barcha elementlarini sikllar ko‘paytmasi shaklida yozing.
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Ixtiyoriy α, β ∈ S_n o‘rin almashtirishlar uchun α⁻¹◦β⁻¹◦α◦β ∈ A_n ekanligini
isbotlang.
12. |A_n| = ⁿ₂^! tenglikni isbotlang.
13. S_n simmetrik gruppada uzunligi r ga teng bo‘lgan turli xil sikllar soni ¹_{r (n−}^n!_r)! ga teng bo‘lishini isbotlang.
14. σ ∈ S_n, n ≥ 2 siklning uzunligi k ga teng bo‘lishi uchun, ord(σ) = k bo‘lishi zarur va yetarli ekanligini isbotlang.