What is the element of largest order

Reference no: EM131002339

1. Why is the permutation representation (of S_n) not irreducible? Hint: You may start by looking-at the generators of S_n.

2. Using the result above and its decomposition into irreducible representations show that χs₄ (ρ_perm (1, 2, 3)) = 1. Hint: this exercise does involve more than one idea and is not the easiest one on this sheet.

3. What is the element of largest order in S₁₂?

4. Show that the character table on the left below can be completed to the form given on the right. Give a brief and concise explanation of each step.
x ≡ (-1 + i√3)/2: y = (-1 + i√7)/2.

	C₁	C₂	C₃	C₄	C₅
\|C_i\|	1		3	7	7
χ(ρ₂)	1		1	x	x*
χ(ρ₄)		y		0	0

	C₁	C₂	C₃	C₄	C₅
\|C_i\|	1	3	3	7	7
χ(ρ₁)	1	1	1	1	1
χ(ρ₂)	1	1	1	x	x*
χ(ρ₃)	1	1	1	x*	x*
χ(ρ₄)	3	y	y*	0	0
χ(ρ₅)	3	y*	y*	0	0

Reference no: EM131002339

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