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I was reading a paper, related to group theory. I came across two doubts (may be simple, but puzzling me):


Let $G$ be a finite group and consider the lattice of subgroups of $G$. On this lattice, we define a Möbius function $\mu$ by $\sum_{H\geq K} {\mu(H)=\delta_{K,G}}$.

I want two verifications:

1) If $G=C_n$ is cyclic group of order $n$, and $H=C_m$ is any subgroup of order $m$, then $\mu(C_m)=\mu(n/m)$, the Möbius function in number theory.

2) If $G=D_n$, the dihedral group of order $2n$, then for each $m$, $m|n$, consider the subgroups $C_m$ (cyclic group of order $m$), and $D_m$, the dihedral group of order $m$. The valules of $\mu(H)$ for these subgroups are given as:

$\mu(C_m)=-\frac{n}{m}\mu(\frac{n}{m})$.

$\mu(D_m)=\mu(n/m)$ (for both cases of $m$ -odd or even)


(Here the $\mu$ on R.H.S. denotes the Möbius function of number theory).

I tried to verify these, but confused too. Please help. (This question can not be posted on MO, so posted lastly here).

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1  
Duplicate of math.stackexchange.com/questions/25975/… and 20% accept rate. –  Jack Schmidt Mar 13 '11 at 9:50
    
@Jack: I am not satisfied with that answer, and so posted again. (I can not post this question for MO, o.w. I would get reply "Post on MathStackexchange..."...so thats I have done...and I would like to clear the doubts to read the paper further...) –  user8186 Mar 13 '11 at 9:54
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I think it would have been appropriate to mention that this is a follow-up / partial repeat of a previous question. BTW, why are you posting from two different accounts? –  joriki Mar 13 '11 at 10:58
    
@joriki: I am sorry! My previous account is not working (therefore, i made new account with slight modification to user name). Can you help, for this question? I am not proceeding further in the reading of a paper by P. Hall. –  user8186 Mar 13 '11 at 11:01
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The paper is probably P. Hall's Eulerian functions of a group Quart. J. Math. 7 (1936) 134–151. dx.doi.org/10.1093/qmath/os-7.1.134 –  Jack Schmidt Mar 14 '11 at 3:20

2 Answers 2

up vote 2 down vote accepted

Be careful that μ(H) can mean different things in the literature. I'll use the more specific μ(H,G) to indicate the Möbius function for the interval between H and G in the subgroup lattice of G. Philip Hall uses μ(H) for μ(H,G) in his paper:

Hall, P. "The Eulerian functions of a group." Quart. J. Math., Oxf. Ser. 7, 134-151 (1936). JFM:62.0082.02 DOI:10.1093/qmath/os-7.1.134

In order to calculate μ(H,G), I first calculate μ(1,G). This will allow me to use some nice reductions.


My copy of Hall is at the office, but I remembered the paper I was reading and it appears to be perfect for getting started with Möbius functions on groups (especially with regards to getting into the later rep theory):

Hawkes, T.; Isaacs, I. M.; Özaydin, M. "On the Möbius function of a finite group." Rocky Mountain J. Math. 19 (1989), no. 4, 1003–1034. MR1039540 DOI:10.1216/RMJ-1989-19-4-1003

This gives a remarkably useful reduction formula (Cor 3.3) for:

μ(1,G) = −k⋅μ(1,G/A)

where A is any abelian minimal normal subgroup of G with exactly k complements.

This formula (in its Eulerian form) is due to:

Gaschütz, Wolfgang. "Die Eulersche Funktion endlicher auflösbarer Gruppen." Illinois J. Math. 3 (1959) 469–476. MR107670 URL:euclid.ijm/1255455454

Using this reduction formula trivializes the cyclic and dihedral groups, along with the following observation on p. 1010 of Hawkes, et al. The Möbius function μ(1,H) does not depend on the embedding of H in G, it is an invariant of the isomorphism type of the group H. In particular, we can forget about the surrounding group G and just think about the cyclic group and the dihedral group (rather than cyclic subgroups or dihedral subgroups).

If G is a cyclic group of non-square-free order, then it has a cyclic Sylow p-subgroup P that is not order p. The only minimal normal subgroup of G contained in P is Ω(P), the subgroup of G consisting of the elements of order dividing p. This subgroup has no complements at all (just like 2Z/4Z is not complemented in Z/4Z, because Z/4Z has only a single involution). Hence k = 0 and μ(1,G) = −0⋅μ(1,G/Ω(P)) = 0. The normal Möbius function of an integer is 0 if the integer is not square-free.

If G is cyclic of square-free order, then each of its Sylow p-subgroups is order p, and so a minimal normal subgroup P. Each such subgroup P has exactly one complement (the Hall P′-subgroup of G consisting of those elements of order coprime to p). Hence k = 1 and we have μ(1,G) = −μ(1,G/P). Since G/P is a cyclic group of square-free order with one less prime factor we get the formula: μ(1,G) = (−1)j where |G|=n is a square-free number with exactly j prime factors. This is exactly the formula for μ(n).

Now if G is dihedral of order 2n, then μ(1,G) = −n⋅μ(n), again by considering a minimal normal subgroup of order p, which has p complements, unless p2 divides n.


Now we handle the case of μ(H,G). If N is a normal subgroup of G contained within H, then μ(H,G) = μ(H/N,G/N) by Noether's lattice isomorphism theorem.

This handles the cyclic groups immediately, since H=N is normal, we just get μ(H,G) = μ(1,G/H) = μ([G:H]).

This also handles the cyclic subgroups (of rotations) in a dihedral group: if G is dihedral and H ≤ [G,G] then H is cyclic and normal in G, with a dihedral quotient (possibly of order 2 or 4). Hence μ(H,G) = μ(1,G/H) = −kμ(k), where [G:H]=2k.

If H is a dihedral subgroup of a dihedral group G, then N=H∩[G,G] is cyclic, [H:N]=2 and N is normal in G (N is the group of rotations in H). μ(H,G) = μ(H/N,G/N), and so we are left with the case that G is dihedral and H is a reflection (order 2 subgroup not contained within the group of rotations). In this case, H is contained only in dihedral subgroups, one for each divisor of n. In other words, the subgroup lattice from H to G is isomorphic to the divisor lattice of [G:H]. Hence μ(H,G) = μ([G:H]).

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And your (Rahul) usage of μ agrees with Hall, just not with the modern stuff I've seen. –  Jack Schmidt Mar 14 '11 at 4:48
    
Thanks!! –  user8186 Mar 14 '11 at 5:09

you can define moebius functions for posets (http://en.wikipedia.org/wiki/Incidence_algebra). As for the cyclic group, the answer is yes since the lattice of subgroups (under inclusion) is isomorphic to the lattice of divisors of the order of the group. there are also some closed formulas for some groups (for example $p$-groups as in the wikipedia article)

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Thanks for pointing out the p-group formula. I had seen the solvable "formulas" before, but did not understand them. –  Jack Schmidt Mar 9 '11 at 22:14
    
@ Its OK for cyclic groups. Also, I would like to get an explicit formula for $\mu (H)$, if possible, for any finite group $G$. –  user8186 Mar 11 '11 at 3:06
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Rahul, why do you think there should be an explicit formula for the Mobius function on any finite group? In that generality there might not be any formula other than the definition of it. –  KCd Mar 13 '11 at 18:44
    
@KCd I am doing computations in groups and their characters. The first groups I had chosen for computations are Cyclic, and Dihedral. The values of Mobius functions on such groups can be given in simpler form, and is useful in computation, their characters are well known, and interesting - we have to just look at them geometrically!! But I stuck in finding the values of Mobius function on these groups! –  user8186 Mar 14 '11 at 3:36

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