Chapter 4 – Schur Multiplier, Projective Representations and Characters
PROOF. See Karpilovsky [43].
Theorem 4.1.3. LetGbe a finite group andH be a subgroup of indexm.Then(M(G))m ∼=M(H), where (M(G))m is the group of allmth powers of M(G).
PROOF. See Karpilovsky [43].
Schur [68] reduced the problem of computingM(G) to compute the Schur multiplier of the Sylow p−subgroups ofG,wherepis a prime dividing the order ofG.The following theorem describes the Schur multiplier of a groupG in terms of the Sylowp−subgroups ofG.
Theorem 4.1.4 (Schur [68]). Let S be a Sylow p−subgroup of G. Then the Sylow p-subgroup of M(G) is isomorphic to a subgroup of M(S).
PROOF. See Karpilovsky [43].
Theorem 4.1.5. A group G has trivial Schur multiplier if and only if it has a set of subgroups with trivial Schur multipliers and relatively prime indices.
PROOF. See Karpilovsky [43].
In GAP [30] and Magma [16] there are various commands to calculate the Schur multiplier of a finite groupG.For example the commands “AbelianInvariantsMultiplier(G)” and “pMultiplicator(G,p)”
of GAP and Magma respectively reveals the Schur multiplier of G and of a Sylowp−subgroup of Grespectively. We deal with some examples in Chapters 7 and 8.
Remark 4.2.1. From Definition 4.2.1, we observe that if α(x, y) = 1F, ∀x, y∈G, then we obtain that P(xy) =P(x)P(y) and hence P becomes an ordinary representation of G. Sometimes a pair (P, α) is used to indicate a projective representationP and its associated factor setα.
Definition 4.2.2. A projective representation P : G −→ GL(n,F) is said to be irreducible if there are no non-trivial subspaces of the vector space V(n,F) which are sent into themselves by all the transformations P(g), g∈G.
Now we look at projective representations from different point of view. It is an elementary result (see Basheer [7] for example) that the center Z(GL(n,F)) consists of {λIn| λ ∈ F∗} and the quotientGL(n,F)/Z(GL(n,F)) is called theprojective general linear group, which is denoted byP GL(n,F).Letπ:GL(n,F)−→P GL(n,F) be the natural epimorphism. IfP :G−→GL(n,F) is a projective representation of a group G, then π◦P is a homomorphism ofG into P GL(n,F).
Conversely if ρ is any homomorphism from G into P GL(n,F) and if for each g ∈ G we select a unique element P(g) in the coset π(g) of Z(GL(n,F)) in GL(n,F), choosing P(1G) = In, then g7→P(g) is a projective representation of GoverV(n,F).
G π◦P //
P
P GL(n,F)
GL(n,F)
π
::t
tt tt tt tt tt tt tt tt tt t
Thus the projective F-representations of G can be identified with the homomorphisms of G into the projective general linear group.
We would like to remark that the projective representations occur in an un-avoidable way in the study of ordinary representations.
Theorem 4.2.1. Let N CG and suppose that χ is an irreducible C−representation of N whose character is invariant in G. Then there exists a projective C−representation X of G such that for alln∈N and g∈G we have
(i) X(n) =χ(n), (ii) X(ng) =X(n)X(g),
Chapter 4 – Schur Multiplier, Projective Representations and Characters
(iii) X(gn) =X(g)X(n).
Furthermore, if X0 is another projective representation satisfying (i), (ii) and (iii), then X0(g) = X(g)µ(g) for some function µ:G−→C∗,which is constant on cosets of N.
PROOF. See Isaacs [38].
We now consider the associated factor sets with the projective representations.
Lemma 4.2.2. Let α be the associated factor set with a projective representation P of G. Thenα satisfiesα(xy, z)α(x, y) =α(x, yz)α(y, z) for allx, y, z ∈G.
PROOF. By associativity we have
P(x)P(y)P(z) =α(x, y)P(xy)P(z) =α(x, y)α(xy, z)P(xyz) and
P(x)P(y)P(z) =α(y, z)P(x)P(yz) =α(y, z)α(x, yz)P(xyz).
Now the result follows since P(xyz) is invertible.
As with ordinary representations, we now define equivalence of projective representations. From now on, we will only be concerned with projective representations over the complex fieldC. Definition 4.2.3. Two projective representations P1 and P2 of G are equivalent if there is a non-singular matrix T such that for all g ∈ G, P1(g) = d(g)T P2(g)T−1 for some d(g) ∈ C∗. If d(g) = 1 for all g∈G then P1 and P2 are linearly equivalent.
Having defined what is meant by linearly equivalence of projective representations, another formu- lation to Definition 4.2.2 can be given. A projective representation P is irreducible if it is not linearly equivalent to a projective representation of the form
? ? 0 ?
.
Lemma 4.2.3. If two projective representations are equivalent, then they have equivalent factor sets; if they are linearly equivalent they have equal factor sets.
PROOF. Let (P1, α1) and (P2, α2) be two equivalent projective representations ofG. SupposeT is a non-singular matrix and d : G−→ C∗ such that P1(g) = d(g)T P2(g)T−1 for all g ∈ G. Now for g, h∈G,
α1(g, h) = P1(g)P1(h)(P1(gh))−1
= d(g)T P2(g)T−1d(h)T P2(h)T−1(d(gh))−1T(P2(gh))−1T−1
= d(g)d(h)(d(gh))−1T P2(g)P2(h)(P2(gh))−1T−1
= d(g)d(h)(d(gh))−1α2(g, h),
soα1 and α2 are equivalent. IfP1 and P2 are linearly equivalent, thend(g) = 1 for allg∈Gin the
above expressions, soα1=α2.
Let F[G,C] be the set of all functions λ:G−→ C. If P is a projective representation of G with factor set α and λ∈F[G,C], then P0 =λP, whereP0(g) =λ(g)P(g) for all g∈G, is a projective representation of Gwith factor set α0, and α0 satisfies the relation
α0(x, y) =λ(x)λ(y)(λ(xy))−1α(x, y) ∀x, y∈G. (4.3) Remark 4.2.2. It follows from Eq. (4.3) that α ∼1 if and only if there exists λ∈F[G,C] such that
α(x, y) =λ(x)λ(y)(λ(xy))−1 ∀ x, y∈G.
The following result provides a close connection between the degrees of the irreducible projective characters with factor set α and theo([α]).
Lemma 4.2.4. Berkovich and Zhmud [15]Let (P, α) be a projective representation ofGsuch that deg(P) =n. If o([α]) =m thenm|n.
PROOF. We know that
P(x)P(y) =α(x, y)P(xy).
Taking the determinant for both sides we obtain
det(P(x)) det(P(y)) = det(α(x, y)P(xy))
= α(x, y)ndet(P(xy)) which implies
α(x, y)n= det(P(x)) det(P(y))(det(P(xy))−1.
Chapter 4 – Schur Multiplier, Projective Representations and Characters
By Remark 4.2.2 we obtain [α]n= 1. Hence m|n.
As it has been mentioned in Ali [1] that projective representations of a group G can be obtained by three different ways. In this thesis we are only concerned with one method in which projective representations of a groupGcan be constructed from the ordinary representations of the so-called representation group or covering group. For the other methods one can refer to Ali [1] and more results on the construction of projective representations, by combining the three methods, can be found in a series of articles by Morris ([52], [53], [54]) and Read ([59], [60], [61]).
Definition 4.2.4. A central extension of G is a group H together with a homomorphism ν of H onto G such thatkerν ≤Z(H).
Now let ν : C(G) −→ G be an epimorphism with kernel N and suppose that C(G) is a central extension of N by G. Let τ : G −→ P GL(n,C) be a projective representation. If there is a homomorphism eτ renders the commutativity of the diagram
C(G) ν //
eτ
G
τ
GL(n,C) π //P GL(n,C)
(i.e., τ ◦ν =π◦τe), then τ is called has the property to be lifted to C(G).We say C(G) has the projective lifting property if every projective representation ofG can be lifted toC(G).
Remark 4.2.3. Note that the homomorphism τe in above commutative diagram is an ordinary representation of C(G) (i.e., a representation over the complexC).
Definition 4.2.5. If G is a group, then a cover or representation group of G is a central extension C(G) of N (for some abelian group N) with the projective lifting property and with N ≤C(G)0.
Schur proved (see Rotman [64]) that every finite groupGhas a cover C(G) withG∼=C(G)/M such that every projective representation ofGdetermines an ordinary representation ofC(G); moreover, it turns out thatM ∼=M(G) andC(G) is the central extension ofM(G) byG.In fact the following theorem, due to Schur, shows the existence of covering groups of any finite groups.
Theorem 4.2.5 (Schur 1904). Every finite group G has a cover C(G), which is the central extension of M(G) by G.
PROOF. See Theorem 7.66 of Rotman [64].
The above theorem affirms that every finite groups G, of order n, has at least one representation group C(G) of order mn where m =|M(G)| and the kernel of the extension is isomorphic to the Schur multiplier M(G) of G.
Definition 4.2.6. LetG1andG2 be any two groups (not necessarily finite). We sayG1is isoclinic to G2 if the following conditions hold:
(i) G1/Z(G1)∼=G2/Z(G2), (ii) G01∼=G02,
(iii) If θ and ψ are the isomorphisms satisfying conditions (i) and (ii) respectively and if for all a1, b1 ∈G1 and a2, b2∈G2 we have
θ(a1Z(G1)) = a2Z(G2), θ(b1Z(G1)) = b2Z(G2), thenψ(a−11 b−11 a1b1) =a−12 b−12 a2b2.
Example 4.2.1. • All the abelian groups are isoclinic, as if we considered any two abelian groups, they will satisfy the conditions in Definition 4.2.6
• Any two isomorphic groups are isoclinic, but the converse is not true, that is not any two isoclinic groups are isomorphic.
Remark 4.2.4. For a finite group G, the covering groups need not be unique, up to isomor- phism, but are unique up to isoclinism. For example (see Rotman [64]) both D8 and Q8 are (non-isomorphic) covers ofZ2×Z2 ∼= 22.Note that one can verify the conditions of Definition 4.2.6 to check that the two groupsD8 andQ8 are isoclinic.
Lemma 4.2.6. Let (H, π) be a central extension of G with A = ker(π). Let X be a set of coset representatives for A inH, and write X ={xg :g∈G},where π(xg) =g. Defineα:G×G−→A by xgxh=α(g, h)xgh.Thenα is anA-factor set of Gand the equivalence class ofα is independent of the choice of X.
PROOF. See Issacs [38].
Chapter 4 – Schur Multiplier, Projective Representations and Characters
Corollary 4.2.7. Let H be a central extension ofG with A, X and α be as in Lemma 4.2.6. Let T be an ordinary representation of H such that the restriction TA is the scalar representation λI for some λ∈Hom(A,C∗),that is
T(a) =
λ(a)
λ(a) . ..
λ(a)
n×n
∀a∈A,
where n= deg(T). Define P(g) =T(xg) for g ∈G. Then (P, λ(α)) is a projective representation of G, where λ(α)(g, h) = λ(α(g, h)). Furthermore, P is irreducible if and only if T is and the equivalence class ofP is independent of the choice of coset representatives X.
PROOF. See Isaacs [38].
Remark 4.2.5. Note that ifT is an ordinary irreducible representation of H then the condition that TA be scalar representation is satisfied by the Schur’s lemma (see Moori [49]), since A lies in the center of H.