Subdirect Products of Groups
February 13, 2011
In this note we sketch a proof of a theorem about subgroups of direct products of groups, and
describe a consequence for subdirect powers of finite nonabelian simple groups. The theorem is a
slight generalization of a well-known result due to Remak, Klein, and Fricke (cf. Rose [1], Theorem
8.19 and Exercise 439; an excerpt of the relevant pages is available online1 ).
Let T1 , T2 , . . . , Tn be a collection of groups and suppose X is a subgroup of their direct product:
X6
Y
Tk = T1 × T2 × · · · × Tn .
Q
Q
Tk denote the
Tk → Ti be the usual projection epimorphism, and let π̂i : Tk →
k6=i
Q
Q
projection of Tk onto the “complement” of Ti , which we will denote by T̂i =
Tk .
Let πi :
Q
k6=i
Theorem 1. If X, Ti , and T̂i are as above, then for all i = 1, . . . , n,
(i) Ti ∩ X P πi (X) 6 Ti
and
T̂i ∩ X P π̂i (X) 6 T̂i ,
(ii) (T1 ∩ X) × · · · × (Tn ∩ X) P X, and
(iii) πi (X)/(Ti ∩ X) ∼
= X/ (Ti ∩ X) × (T̂i ∩ X) ∼
= π̂i (X)/(T̂i ∩ X).
Proof sketch: (coming soon) The case n = 2 is illustrated in the diagram in Figure 1.
T1 × T2
X
T2
T1
π2 (X)
π1 (X)
T2 ∩ X
T1 ∩ X
(T1 ∩ X) × (T2 ∩ X)
(e)
Figure 1: Illustrates Theorem 1 in case n = 2. Solid lines represent subgroup relations. Intervals
colored red are isomorphic. Dashed lines emphasize the fact that the projections, π1 (X) and π2 (X),
are not generally subgroups of X.
1
http://www.math.hawaii.edu/~williamdemeo/groups/Rose-ExcerptOnDirectProducts.pdf
1
Subdirect Products of Groups
February 13, 2011
Recall, X is a subdirect product of T1 , . . . , Tn provided X 6
Q
Tk and the projections are onto:
k
sd
πi (X) = Ti . We denote this situation by X 6
Q
Tk . The following is a consequence of Theorem 1:
sd
Corollary 1. If T is a finite nonabelian simple group and X 6 T n , then T ∼
= X/(T n−1 ∩ X).
Thus, there are (at least) n distinct normal subgroups of X (namely, T̂ ∩ X) with the property that
the quotient group is isomorphic to T . In case n = 2, every subdirect product of T 2 is isomorphic
to T .
Proof: (coming soon)
References
[1] John S. Rose. A Course on Group Theory. Dover, New York, 1978.
2