1.4. The group defined by the permutations of four objects, P(4), is isomor-phic (has a one-to-one correspondence) with the group of symmetry opera-tions of a regular tetrahedron (Ta). The symmetry operations of this groupare sufficiently complex so that the power of group theoretical methods can beappreciated. For notational convenience, the elements of this group are listedbelow.= (4213)t = (4231)e%3 (1234) д %3 (3124) т %3 (1423)n = (1432)(4123)p= (4132)q = (2413) w =(2431) y =a = (1243) h = (3142)b = (2134) i = (2314)c = (2143) j= (2341)(3412)(3421)(4312)0 == nv =%3|d = (1324) k = (3214)f = (1342) 1= (3241)(4321). We will now consider three irreducible representations for the permutationgroup P(3):EABI :(1)(1)(1)(-1)(1)(-1)(6?) (71) (.EG

Answered: Image /qna-images/answer/3247c2db-6163-4f44-bd87-514745753a06.jpg

We will now consider three irreducible representations for the permutation group P(3): E A B I : (1) (1) (1) (-1) (1) (-1) C) () I2 : EG sh. (2.70) (2.5) F In : (1) (-1) (1) (1) (1) (1) 2.2. Consider the example of the group P(3) which has six elements. Using the irreducible representations of Sect. 2.3, find the sum of l?. Does the equality or inequality in (2.70) hold? Can P(3) have an irreducible representation with l; = 3? Group P(4) has 24 elements and 5 irreducible representations. Using (2.70) as an equality, what are the dimensionalities of these 5 irreducible representations (see Problem 1.4)?

We will now consider three irreducible representations for the permutation group P(3): E A B I : (1) (1) (1) (-1) (1) (-1) C) () I2 : EG sh. (2.70) (2.5) F In : (1) (-1) (1) (1) (1) (1) 2.2. Consider the example of the group P(3) which has six elements. Using the irreducible representations of Sect. 2.3, find the sum of l?. Does the equality or inequality in (2.70) hold? Can P(3) have an irreducible representation with l; = 3? Group P(4) has 24 elements and 5 irreducible representations. Using (2.70) as an equality, what are the dimensionalities of these 5 irreducible representations (see Problem 1.4)?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...

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