2013 final exam in Group Theory(A)s (1)

G , the identity element, indeed
(2) In the most cases, f is the group element e of
r ( e) dim(V )
for any representation of f is the group element e of
G . However, there are some
G.
exceptions, in particular, in the trivial representation, f can be any group element of
(3) The character is constant for the group elements inner the same conjugacy class, and thus, for the group elements, a and b , belong to the same conjugacy class
r?
(2) If r ( f ) dim(V ), f G , what is your identification about the group element f ?
(3) If the group elements of G , a and b , belong to the same conjugacy class, what is the relationship between the two characters, r (a ) and r (b) ? Give the argument for your answer.
U
be representation of
G , and r V be representation of H .
(1) What is the direct product of the two groups, G H ? (2) What is the tensor product of the two vector spaces, U V ? (3) What is the direct-product representation r r of
is defined as the set
1 (eH ) {g | g G, ( g ) eH H } G
where is the identity element of H . (2) For arbitrary two elements a , b G
(a ) (b) (dad 1 )(dbd 1 ) d (ab)d 1 (ab)
r ( a ) r ( cbc 1 ) r ( c ) r (b) r ( c 1 ) r ( c ) r (b) r 1 ( c) r (b)
4. Let G and H be groups, and U and V be vector spaces. Let r
two-dimensional representations of S3 , by using the result in (2). Suggested solution: (1) According to Schur’s Lemma, if r1
not equivalent to

r2 , T is a zero-operator;
not equivalent to
a G .

r2 , is T a zero-operator or not?
For what condition of T , the representations r1 and r2 are equivalent to each other, r1 r2 ? (2) Find the matrix elements of the group elements (23) and (123) in the natural two-dimensional representations of S3 . (3) Prove that (132) (123)2 , find the matrix elements of the group element (132) in the natural
r[(123)] 1/ 2 3 / 2 . 3 / 2 1/ 2
(3) From
(123)2 [123] (123)[231] [312]
where [123] is the three vertices acted by (123)2 , we have an identity for the two 3-cycles
: G G, a (a ) dad 1 , for given d and a G
is a homomorphism (called an automorphism). (3) Write down the kernel of the mapping from the group G onto itself defined in (2). Suggested solution: (1) The kernel of the mapping
Therefore, the map from a group G onto itself is a homomorphism. (3) The kernel of the mapping is
1 (e) g | g G, ( g ) e G
g | g G, dgd 1 e G g | g G, g d 1ed e G e
r ( a ) r ( b) ,
Indeed, because a is conjugate to b in
a cbc 1
G , there exists an element c of G , such that
and a direct manipulation leads to
1 3 1/ 2 ; ˆy ˆx ˆx e ˆ r[(123)]e e e x 2 2 3 / 2 3 1 3 / 2 ˆy e ˆ ˆx e ˆy r[(123)]e e . y 2 2 1/ 2
Therefore, the corresponding representation matrix is of the form
be a mapping from the group G onto H
:G H g (g)
What is the concept of the kernel of a mapping
from the group G onto H .
(2) Prove that the map from a group G onto itself
Suggested solution:
(1) A series
r a series r ( a ) trr (a ), r (b) trr ( c ), ; a, b, c, G
is called a character of the representation
r.
we have
1/ 2 3 / 2 1/ 2 r (132) 3 / 2 1/ 2 3 / 2
2
3 / 2 1/ 2
3. (1) A group G {a, b, c,} acts on a vector space V to give a representation r . What is the character of the representation
ˆx and e ˆy (directed to the vertex 1 of the triangle), these origin, and the Cartesian basis of 2 as e
bases span a natural two-dimensional representation of S3 , 2 . Then a reflection (23) (interchange the vertices 2 and 3 of the triangle) has the following action on the two bases in 2
123 (123)2 (132) . 312
Using
r[(123)] 1/ 2 3 / 2 3 / 2 1/ 2
and homomorphism version of (132) (123)2
r (132) [ r (123)]2
ˆ y 1 ; ˆx e ˆ x 0e r[(23)]e 0 ˆx e ˆy 0 . ˆ y 0e r[(23)]e 1
so that the corresponding representation matrix is of the form

r[(23)] 1 0 . 0 1
U V
G H on U V ?
(4) Prove that r r
U
V

is really a representation of G H on U V .
Suggested solution:
(1) Their direct product G H consists of all pairs, ( a G, b H ) , with the multiplication
The final examination in Group Theory School of Physics and Technology, Wuhan University, Winter semester, 2013. 1. (1) Suppose that G and H be two groups, and (A-version)