Character board

A character table contains information about the irreducible representations of a finite group . In chemistry , they can be used to make statements about the properties of molecules based on the associated point group .

The actual character table of a group is a square table with complex numbers as entries. The lines correspond to the irreducible representations of , the columns to the conjugation classes in . The table entry for representing and conjugation is to the value of the associated character , evaluated on any element of . ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \rho }$ ${\displaystyle C}$ ${\displaystyle \rho }$ ${\displaystyle C}$

Definitions

Every irreducible representation of a finite group in the group of invertible n × n matrices defines the associated irreducible character ${\displaystyle \rho \colon G\rightarrow \mathrm {GL} _{n}(\mathbb {C} )}$ ${\displaystyle G}$

${\displaystyle \chi _{\rho }\colon G\rightarrow \mathbb {C} ,\,\chi _{\rho }(g):=\mathrm {tr} (\rho (g))}$,

where is the track map . Two irreducible representations are equivalent if and only if the associated irreducible characters are the same. Are conjugate , then is for a and therefore follows the traces for a character according to the properties ${\displaystyle \mathrm {tr} }$ ${\displaystyle g,h\in G}$ ${\displaystyle g=k^{-1}hk}$ ${\displaystyle k\in G}$ ${\displaystyle \chi =\chi _{\rho }}$

${\displaystyle \chi (g)=\mathrm {tr} (\rho (g))=\mathrm {tr} (\rho (k^{-1}hk))=\mathrm {tr} (\rho (k)^{-1}\rho (h)\rho (k))=\mathrm {tr} (\rho (h))=\chi (h)}$,

that is, characters are constant on conjugation classes . Therefore a character is already determined by the fact that the value is given on all conjugation classes. It can also be shown that there are as many irreducible characters as there are conjugation classes. Therefore one can describe all characters by a square scheme. The columns of this scheme are the conjugation classes , the rows are the characters . The i-th row and j-th column contain the value of on the conjugation class , that is , where a representative element of the conjugation class is selected with. ${\displaystyle C_{1},\ldots ,C_{r}}$ ${\displaystyle \chi _{1},\ldots ,\chi _{r}}$ ${\displaystyle \chi _{i}}$ ${\displaystyle C_{j}}$ ${\displaystyle \chi _{i}(c_{j})}$ ${\displaystyle c_{j}\in C_{j}}$

Among the characters there is an excellent element, namely the character for trivial representation, which assumes the value 1 in all conjugation classes, which is also called the trivial character. There is also a trivial conjugation class that consists of the neutral element 1. The value of each character on the trivial conjugation class is the trace of the identity matrix and thus equal to the dimension of the i-th irreducible representation. ${\displaystyle d_{i}}$

You now arrange the characters so that the trivial character and the trivial conjugation class is. There is no arrangement for the other data; many authors choose the first column in ascending order. The column labeling consists of the conjugation class or a representative element, often the cardinality of the conjugation class is also given. This leads to the following overview, which is called the character board: ${\displaystyle \chi _{1}}$ ${\displaystyle C_{1}=\{1\}}$

${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle |C_{2}|}$	${\displaystyle \ldots }$	${\displaystyle |C_{r}|}$
	${\displaystyle 1}$	${\displaystyle c_{2}}$	${\displaystyle \ldots }$	${\displaystyle c_{r}}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \ldots }$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle d_{2}}$	${\displaystyle \chi _{2}(c_{2})}$	${\displaystyle \ldots }$	${\displaystyle \chi _{2}(c_{r})}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \ddots }$	${\displaystyle \vdots }$
${\displaystyle \chi _{r}}$	${\displaystyle d_{r}}$	${\displaystyle \chi _{r}(c_{2})}$	${\displaystyle \ldots }$	${\displaystyle \chi _{r}(c_{r})}$

This character board contains important group data. It is true that the group cannot be reconstructed from this, but it contains enough information to decide important properties of the group. Often it is possible to determine data on the character board without knowing the group exactly. Various sentences serve this purpose, such as the fact that

${\displaystyle 1+d_{2}^{2}+d_{3}^{2}+\ldots +d_{r}^{2}=|G|}$

and above all the Schursch orthogonality relations .

Examples

Cyclic groups

Let it be the cyclic group with elements. The elements are designated with . Be . Then you get the following character board: ${\displaystyle G=\mathbb {Z} _{r}}$ ${\displaystyle r}$ ${\displaystyle [0],[1],\ldots [r-1]}$ ${\displaystyle \omega =e^{2\pi i/r}\in \mathbb {C} }$

${\displaystyle \mathbb {Z} _{r}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \ldots }$	${\displaystyle 1}$
	${\displaystyle [0]}$	${\displaystyle [1]}$	${\displaystyle [2]}$	${\displaystyle \ldots }$	${\displaystyle [r-1]}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \ldots }$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle \omega }$	${\displaystyle \omega ^{2}}$	${\displaystyle \ldots }$	${\displaystyle \omega ^{r-1}}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle \omega ^{2}}$	${\displaystyle \omega ^{4}}$	${\displaystyle \ldots }$	${\displaystyle \omega ^{r-2}}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \ddots }$	${\displaystyle \vdots }$
${\displaystyle \chi _{r}}$	${\displaystyle 1}$	${\displaystyle \omega ^{r-1}}$	${\displaystyle \omega ^{r-2}}$	${\displaystyle \ldots }$	${\displaystyle \omega }$

It is . ${\displaystyle \chi _{i}([j])=\omega ^{(i-1)j}}$

Klein group of four

It is the small group of four . Since and since one can show that the characters of a direct product of groups are the products of the characters of the factors of this product, the character table can be obtained from that of . You get: ${\displaystyle V_{4}=\{1,a,b,ab\}}$ ${\displaystyle V_{4}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{2}}$

${\displaystyle V_{4}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
	${\displaystyle 1}$	${\displaystyle a}$	${\displaystyle b}$	${\displaystyle ab}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$
${\displaystyle \chi _{4}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$

Symmetrical group S ₃

In addition to the trivial homomorphism and the signum function, the symmetrical group S ₃ must have at least one other irreducible character and at least one with a dimension greater than 1, since the group would otherwise be Abelian. Since the sum of the squares of the dimensions is equal to the group order, only the other two values ​​of are fixed by means of the orthogonality relations . You get: ${\displaystyle \chi _{3}}$ ${\displaystyle d_{3}=2}$ ${\displaystyle \chi _{3}}$

${\displaystyle S_{3}}$	${\displaystyle 1}$	${\displaystyle 3}$	${\displaystyle 2}$
	${\displaystyle 1}$	${\displaystyle (1,2)}$	${\displaystyle (1,2,3)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{3}}$	${\displaystyle 2}$	${\displaystyle 0}$	${\displaystyle -1}$

Non-Abelian 8-element group

Just from the knowledge that a group has 8 elements and is not Abelian, the character table can be constructed. Since is a 2 group , the center cannot be trivial and has 2 or 4 elements and cannot be cyclic, otherwise it would be Abelian. So must be. Since there can be at most irreducible characters of dimension 2, since with the trivial character there is always one of dimension 1 and since the sum of the squares of dimensions must add up to 8, there is only the possibility of 5 irreducible characters of dimensions 1.1, 1,1,2. So there must also be 5 conjugation classes whose respective representatives are denoted by, where . Means , and orthogonality can be shown that necessary following character table must be present: ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle Z}$ ${\displaystyle G/Z}$ ${\displaystyle G}$ ${\displaystyle G/Z\cong V_{4}}$ ${\displaystyle 3^{2}=9>8}$ ${\displaystyle 1,-1,a,b,c}$ ${\displaystyle -1\in Z}$ ${\displaystyle |Z|=2}$ ${\displaystyle G/Z\cong V_{4}}$

${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 2}$
	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle a}$	${\displaystyle b}$	${\displaystyle c}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$
${\displaystyle \chi _{4}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{5}}$	${\displaystyle 2}$	${\displaystyle -2}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$

Since there are two non-isomorphic nonabelian groups of order 8 with the dihedral group D ₄ and the quaternion group , this example shows that the group cannot be reconstructed from the character table.

Alternating group A ₄

The alternating group A ₄ is non-Abelian and has 4 conjugation classes. For the dimensions of the presentation spaces then only the sequence 1,1,1,3 remains and it results

${\displaystyle A_{4}}$	${\displaystyle 1}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 4}$
	${\displaystyle 1}$	${\displaystyle (1,2)\,(3,4)}$	${\displaystyle (1,2,3)}$	${\displaystyle (1,3,2)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \textstyle e^{\frac {2\pi i}{3}}}$	${\displaystyle \textstyle e^{\frac {4\pi i}{3}}}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle \textstyle e^{\frac {4\pi i}{3}}}$	${\displaystyle \textstyle e^{\frac {2\pi i}{3}}}$
${\displaystyle \chi _{4}}$	${\displaystyle 3}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle 0}$

Alternating group A ₅

The considerations for the character table of the alternating group A ₅ are a bit more complicated. Therefore, only the result should be given here:

${\displaystyle A_{5}}$	${\displaystyle 1}$	${\displaystyle 15}$	${\displaystyle 20}$	${\displaystyle 12}$	${\displaystyle 12}$
	${\displaystyle 1}$	${\displaystyle (1,2)\,(3,4)}$	${\displaystyle (1,2,3)}$	${\displaystyle (1,2,3,4,5)}$	${\displaystyle (1,3,5,2,4)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 4}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$
${\displaystyle \chi _{3}}$	${\displaystyle 5}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle 0}$
${\displaystyle \chi _{4}}$	${\displaystyle 3}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$	${\displaystyle {\frac {1-{\sqrt {5}}}{2}}}$
${\displaystyle \chi _{5}}$	${\displaystyle 3}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle {\frac {1-{\sqrt {5}}}{2}}}$	${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$

Symmetrical group S ₅

Finally, a somewhat larger example should be given with the character table of the symmetrical group : ${\displaystyle S_{5}}$

${\displaystyle S_{5}}$	${\displaystyle 1}$	${\displaystyle 10}$	${\displaystyle 15}$	${\displaystyle 20}$	${\displaystyle 30}$	${\displaystyle 20}$	${\displaystyle 24}$
	${\displaystyle 1}$	${\displaystyle (1,2)}$	${\displaystyle (1,2)\,(3,4)}$	${\displaystyle (1,2,3)}$	${\displaystyle (1,2,3,4)}$	${\displaystyle (1,2,3)\,(4,5)}$	${\displaystyle (1,2,3,4,5)}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{3}}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 0}$	${\displaystyle -1}$	${\displaystyle -1}$
${\displaystyle \chi _{4}}$	${\displaystyle 4}$	${\displaystyle -2}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle -1}$
${\displaystyle \chi _{5}}$	${\displaystyle 5}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle 0}$
${\displaystyle \chi _{6}}$	${\displaystyle 5}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 0}$
${\displaystyle \chi _{7}}$	${\displaystyle 6}$	${\displaystyle 0}$	${\displaystyle -2}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 1}$

Properties of the character board

As the example of the non-Abelian groups of order 8 shows, the group cannot generally be reconstructed from the character table. Nevertheless, certain group characteristics can be read off.

A group is Abelian (commutative) if and only if all irreducible representations are one-dimensional, that is, if the first column of the group table contains only ones.

One can show that for every irreducible character there is a normal divisor and that every other normal divisor is an average of such . ${\displaystyle N_{\chi }=\{g\in G|\chi (g)=\chi (1)\}}$ ${\displaystyle N_{\chi }}$

In particular, a group is simple if and only if in every row from the second the value of the first column in the row (i.e. the dimension of the associated display space) does not appear a second time. It is therefore easy to see that it is easy. The second line of the character board shows that this group is not easy. ${\displaystyle A_{5}}$ ${\displaystyle A_{4}}$

Since one knows the normal divisors and their subset relationships with the, one can also specify methods with which one can read off solvability and, with a little more effort, also nile power . ${\displaystyle N_{\chi }}$

Application in chemistry

If one now deduces the point group of a molecule from the symmetry elements or with the aid of the Schoenflies scheme , one can infer certain properties of the substance with the help of the character table.

example

The character table of the point group (small group of four, here written in the Schoenflies symbolism ) is expanded as follows with chemically relevant information ${\displaystyle C_{2v}}$

${\displaystyle C_{2v}}$	${\displaystyle E}$	${\displaystyle C_{2}}$	${\displaystyle \sigma _{v}(xz)}$	${\displaystyle \sigma _{v}'(yz)}$	${\displaystyle h=4}$
${\displaystyle A_{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle z}$	${\displaystyle x^{2},y^{2},z^{2}}$
${\displaystyle A_{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle R_{z}}$	${\displaystyle xy}$
${\displaystyle B_{1}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle x,R_{y}}$	${\displaystyle xz}$
${\displaystyle B_{2}}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle y,R_{x}}$	${\displaystyle yz}$

The first name is the point group, in the first line are the symmetry elements R that are contained in it. If a symmetry element occurs n times, then you write or add another line with the numbers as above. In this case all numbers are equal to 1. The symmetry elements form a class with the order . The total number of symmetry elements is the order of the group . In the example of the point group , the order is four ( ). The irreducible representations are in the first column . In the example above, these are shown in Latin letters with indices, the so-called Mulliken symbols . The following columns contain the values ​​of the characters (here: −1 and +1). In the last two columns are the bases of the irreducible representations, or orbitals , which are as an irreducible representation transform . One says z. B. the rotation around the z-axis transformed like . ${\displaystyle nR}$ ${\displaystyle nR}$ ${\displaystyle n}$ ${\displaystyle C_{2v}}$ ${\displaystyle h=4}$ ${\displaystyle \Gamma _{i}}$ ${\displaystyle \chi ^{R}}$ ${\displaystyle R_{z}}$ ${\displaystyle A_{2}}$

The penultimate column allows conclusions to be drawn as to whether a molecular movement - characterized in its symmetry by its irreducible representation - can be visible in infrared spectroscopy or an electronic transition between two orbitals in UV / VIS spectroscopy (transformed like , i.e. shows translation - where it each time there is a change in dipole moment). Or whether, as can be seen from the last column, the molecular movement can be detected by means of Raman scattering (transformed as and , i.e. according to the polarizability tensor with an associated change in polarizability). ${\displaystyle \Gamma _{i}}$ ${\displaystyle x,y,z}$ ${\displaystyle xy,xz,yz}$ ${\displaystyle x^{2},y^{2},z^{2},x^{2}-y^{2}}$

Rotations and vibrations

• The indications , and relate to molecular rotations in the x, y and z directions, which transform like the irreducible representations. z. B. transforms in a molecule of the point group the rotation around the z-axis like . ${\displaystyle R_{x}}$ ${\displaystyle R_{y}}$ ${\displaystyle R_{z}}$ ${\displaystyle C_{2v}}$ ${\displaystyle R_{z}}$ ${\displaystyle A_{2}}$

The natural vibrations of the molecule also transform like one of the irreducible representations of the point group of the molecule.

Orbitals

The symmetries of the base orbitals of a molecule can also be assigned to an irreducible representation of the point group. Does a character with a certain representation and a certain symmetry element z. B. the character "+1", then the sign of the wave function does not change when using this symmetry element. If it is "-1" then it changes.

Example (continued)

A molecule belongs to the point group (see character table above). The -orbital, which lies on the -axis and how transformed, belongs to its basic set . Reflection at the mirror plane does not change the orbital, it is mapped onto itself, the character is "+1". On the other hand, if you mirror the -orbital at the -plane, the sign of the wave function changes, so the character is "-1", as can be seen from the character table. ${\displaystyle C_{2v}}$ ${\displaystyle p_{x}}$ ${\displaystyle x}$ ${\displaystyle B_{1}}$ ${\displaystyle xz}$ ${\displaystyle x}$ ${\displaystyle yz}$

Reducible and irreducible representations, reduce

An irreducible representation only has and as invariant subspaces . Mix all other sub-rooms. A reducible representation breaks down into various sub-spaces. ${\displaystyle \Gamma _{\mathrm {red} }}$ ${\displaystyle {0}}$ ${\displaystyle L}$

When a representation is completely reducible it can be viewed as a direct sum of irreducible representations . Not every reducible representation is completely reducible. ${\displaystyle \rho }$

In the case of completely reducible representations, the proportions of the irreducible representations in a reducible representation can be determined using rates or the following formula: ${\displaystyle a_{i}}$

${\displaystyle a_{i}={1 \over h}\sum _{R}n\chi ^{R}\chi _{i}^{R}}$

${\displaystyle h}$is the order of the group, the order of the class, the character of the respective irreducible representation and the character of the reducible representation . ${\displaystyle n}$ ${\displaystyle \chi _{i}^{R}}$ ${\displaystyle \Gamma _{i}}$ ${\displaystyle \chi ^{R}}$ ${\displaystyle \Gamma _{\mathrm {red} }}$

${\displaystyle \Gamma _{\mathrm {red} }=\sum _{R}a_{i}\Gamma _{i}}$

See also

• Infrared spectroscopy
• Raman spectroscopy
• Alternative ban
• Chirality

literature

• JH Conway : Atlas of Finite Groups, Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford 1985, ISBN 0-198-53199-0 .

Individual evidence

1. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, example 1
2. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, example 3
3. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, example 6
4. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , example 9.7.1 b
5. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, example 8
6. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , example 9.7.1 c
7. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, example 9
8. ^ GD James, MW Liebeck: Representations and characters of groups . 2nd Edition. Cambridge University Press (2001), ISBN 0-511-01700-6 , chap. 19, example 16.
9. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, sentence 7
10. ↑ JL Alperin, RB Bell: Groups and Representations , Springer-Verlag (1995), ISBN 0-387-94525-3 , chap. 6, corollaries 9 and 14