# The finite groups with a zero entry in each column of its character table (except the first one)

$$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$$Consider the class of finite groups $$G$$ having a zero entry in each column of its character table (except the first one), i.e. for all $$g \neq e$$ there is an irreducible character $$\chi$$ such that $$\chi(g) = 0$$.

I have been led to consider such character table, here are the examples found (see the GAP codes in Appendix):

• at order less than $$384$$, this class reduces to $$A_5$$, $$S_5$$, $$\PSL(2,7)$$, $$\Aut(\PSL(2,7))$$ and $$A_6$$,
• every simple group of order less than $$3000000$$ is in this class except $$A_7$$, $$M_{22}$$,
• every perfect group of order less than $$3600$$ in this class is simple,
• if $$5 \le n \le 19$$, then $$A_n$$ is not in this class iff $$n \in \{ 7, 11, 13, 15, 16, 18, 19\}$$. Idem for $$S_n$$.

Question: What are the finite groups in this class? Which simple groups are not in?

The group $$\PSL(2,q)$$ is in this class iff it is simple, iff $$q \ge 4$$ (the generic character table is known).

Observe that all the examples found above are almost simple, but:

Proposition: This class is stable by direct product.
proof: Immediate by Theorem 4.21 in Isaacs' Character Theory of Finite Groups stating that the irreducible characters of a direct product are exactly the product of the irreducible characters of the components. $$\square$$

Note that $$A_5 \times A_5$$ is a non almost-simple finite group in this class, maybe the smallest one.

Appendix

Small groups:

gap> for o in [2..383] do n:=NrSmallGroups(o);; for d in [1..n] do G:=SmallGroup(o,d);; L:=Irr(CharacterTable(G));; l:=Length(L);; a:=0;; for j in [2..l] do LL:= List([1..l], i-> L[i][j]);; if not 0 in LL then a:=1;; break; fi; od; if a=0 then Print([G,[o,d]]); fi; od; od;
[ AlternatingGroup( [ 1 .. 5 ] ), [ 60, 5 ] ][ SymmetricGroup( [ 1 .. 5 ] ), [ 120, 34 ] ][ Group( [ (3,4)(5,6), (1,2,3)(4,5,7) ] ), [ 168, 42 ] ][ Group( [ (1,4,6,8,5,2,7,3), (1,3,8,6,5,4,7) ] ), [ 336, 208 ] ][ AlternatingGroup( [ 1 .. 6 ] ), [ 360, 118 ] ]


Simple Groups:

gap> it:=SimpleGroupsIterator(10,3000000);; for G in it do L:=Irr(CharacterTable(G));; l:=Length(L);; a:=0;; for j in [2..l] do LL:= List([1..l], i-> L[i][j]);; if not 0 in LL then a:=1;; break; fi; od; if a=1 then Print([G]); fi; od;
[ A7 ][ M22 ]


Perfect Groups:

gap> for o in [60..3599] do n:=NumberPerfectGroups(o);; for d in [1..n] do G:=PerfectGroup(o,d);; L:=Irr(CharacterTable(G));; l:=Length(L);; a:=0;; for j in [2..l] do LL:= List([1..l], i-> L[i][j]);; if not 0 in LL then a:=1;; break; fi; od; if a=0 then Print([G,[o,d]]); fi; od; od;
[ A5, [ 60, 1 ] ][ L3(2), [ 168, 1 ] ][ A6, [ 360, 1 ] ][ L2(8), [ 504, 1 ] ][ L2(11), [ 660, 1 ] ][ L2(13), [ 1092, 1 ] ][ L2(17), [ 2448, 1 ] ][ L2(19), [ 3420, 1 ] ]


Alternating groups:

gap> for n in [5..19] do s:=Concatenation("A",String(n));;  L:=Irr(CharacterTable(s));; l:=Length(L);; a:=0;; for j in [2..l] do LL:= List([1..l], i-> L[i][j]);; if not 0 in LL then a:=1;; break; fi; od; if a=1 then Print([s]); fi; od;
[ "A7" ][ "A11" ][ "A13" ][ "A15" ][ "A16" ][ "A18" ][ "A19" ]

• Is there an application that makes these groups particularly natural or interesting, or is it just curiosity? Aug 17, 2021 at 13:18
• @LSpice: there is an application in tensor category: a pivotal categorification (over a field possibly of positive characteristic) of the character ring of such a finite group must be pseudo-unitary. Aug 17, 2021 at 13:46
• This doesn’t answer your question, but a lovely old theorem of Burnside says every row has a zero except the ones coming from 1-dimensional reps. Aug 17, 2021 at 17:12
• Sebastien, What happens when the "character" table does not come from a group --- do you still get something pseudo-unitary ? Aug 17, 2021 at 19:29
• @InesInstitoris: yes (if I am not mistaken) a pivotal categorification of a commutative fusion ring having such a character table must be pseudo-unitary. I don't how to prove that without the pivotal assumption. Aug 17, 2021 at 20:02

Partial answer: the finite group $$G$$ is clearly in this class if it has a $$p$$-block of defect zero for every prime $$p$$ which divides $$|G|$$. This is a sufficient condition which may not be necessary. No non-trivial finite solvable group satisfies the sufficient condition. Most (but not all) finite simple groups satisfy the sufficient condition (for example, $$M_{22}$$ and $$M_{24}$$ do not, and many alternating groups do not).

Later edit: In the other direction, if $$G$$ is a finite group such that $$F^{\ast}(G)$$ is a $$p$$-group for some prime $$p$$, then whenever $$S$$ is a Sylow $$p$$-subgroup of $$G$$, and $$z$$ is any non-identity element of $$Z(S)$$, then no irreducible character of $$G$$ vanishes at $$z$$, so $$G$$ does not have the desired property. This is because $$G$$ has only one $$p$$-block, the principal block, and no irreducible character in the principal $$p$$-block vanishes at a central element of a Sylow $$p$$-subgroup of $$G$$ by the fundamental congruence of central characters due to R. Brauer.

Third edit: A well-known paper of Granville and Ono, "Defect zero $$p$$-blocks for finite simple groups", Trans AMS, 348,1, January 1996, gives a complete determination of finite non-Abelian simple groups $$G$$ which do not have a $$p$$-block of defect zero for some prime $$p$$. All other non-Abelian finite simple groups $$G$$ have the property of the question, that is, for each non-identity element $$x \in G$$, there is an irreducible complex character $$\chi$$ of $$G$$ with $$\chi(x) = 0$$.

• I confirm that your sufficient condition is not necessary, but there are just $7$ non-abelian and non-alternating finite simple groups which are both in this class and do not satisfy your sufficient condition: $M_{12}$,$J_2$,$HS$,$Suz$,$Ru$,$C_1$, $BM$ (see my answer showing that every non-abelian and non-alternating finite simple group is in this class except $M_{22}$, $M_{24}$ and $C_3$). We still do not know whether there exists a non almost-simple group in this class. We also do not know whether an almost-simple group is in this class if and only if the corresponding simple group is in. Aug 19, 2021 at 5:50
• I think it is easy to construct examples in the class which are not almost simple. I will edit my answer to give an example. An example $G$ with $F^{\ast}(G)$ not a direct product of non-Abelian simple groups would be interesting. Aug 19, 2021 at 8:09
• You are right, it is easy to construct such examples: $A_5 \times A_5$ is in this class but is not almost simple. Aug 19, 2021 at 8:17
• Yes indeed, so I will not re-edit. Aug 19, 2021 at 8:20
• I wonder whether $A_5 \times A_5$ is the group of smallest order among the non almost-simple groups in this class. Aug 19, 2021 at 9:06

By using the answer of Geoff Robinson, we can prove that every non-abelian and non-alternating finite simple group is in this class except $$M_{22}$$, $$M_{24}$$ and $$C_3$$. The result follows from the following computation together with Corollary 2 in Granville and Ono - Defect zero $$p$$-blocks for finite simple groups.

gap> S:=["M12","M22","M24","J2","HS","Suz","Ru","C1","C3","BM"];;
gap> for s in S do L:=Irr(CharacterTable(s));; l:=Length(L);; a:=0;; for j in [2..l] do LL:= List([1..l], i-> L[i][j]);; if not 0 in LL then a:=1;; break; fi; od; if a=1 then Print([s]); fi; od;
[ "M22" ][ "M24" ][ "C3" ]


Corollary 2. Every finite simple group $$G$$ has a $$p$$-block of defect $$0$$, for every prime $$p$$, except in the following cases:

• $$G$$ has no $$2$$-block of defect $$0$$ if it is isomorphic to $$M_{12}$$, $$M_{22}$$, $$M_{24}$$, $$J_2$$, $$\mathit{HS}$$, $$\mathit{Suz}$$, $$\mathit{Ru}$$, $$C1$$, $$C3$$, $$\mathit{BM}$$, or $$A_n$$ where $$n \ne 2m^2 + m$$ nor $$2m^2 + m + 2$$ for any integer $$m$$.

• $$G$$ has no $$3$$-block of defect $$0$$ if it is isomorphic to $$\mathit{Suz}$$, $$C3$$, or $$A_n$$ with $$3n + 1 = m^2 r$$ where $$r$$ is squarefree and divisible by some prime $$q \equiv 2 \bmod 3$$.