Vol. 8 No 7 - Pi Mu Epsilon

contained in both C1 and C2.
Assume a(Cl fl C2)a1 and b(Cl fl C2)b1. Then, aC1al and aC2a1 and
bC1bl and bC2b1.
(ab)C2(a'b1 1.
relation.
and C2.
Since C and C are congruences, (ab)C1(aW) and
1 2
Thus, (ab)(Cl fl C2)(a1b1), and C1 fl C 2 is a congruence
Let J also be a congruence relation that is contained in both Cl
So, J c Cl and J c C . (a,b) e J + (a,b) e C, and (a,fc) e C2
Thus, Cl n C is the largest such congruence, and C 0 C is the infimum
1 2
Lemma. 2.
The set of all normal subgroups of a group G, with the
partial order of set inclusion, is a lattice.
The proof of Lemma 2 follows that of Lemma 1, and uses standard
group theoretic results. -
Define f: Nor(G) + C dG) by f(N) = C where aC# if and only if .
Na = Nb for all a, b e G.
Themem.
lattice isomorphism.
The mapping f: Nor(G) + Con(G) given by f(S) = C is a
Proof. Assume f(Nl) = f(N2). Then, C = C x E N * Nlx =
1
Now, consider Cl o C2.
Assume a(C1 o C2)d and d(Cl o C2)g.
There exist b, f e G such that aClb and bC2d and dClf and fCg.
aC1b and b d " ~ e and eC1d"f and fC2g, where e is the identity of G.
2
Since Cl and C2 are congruences, a~~(bd-~f) and (bd"f)C2g.
a(C o C )g, and Cl o C2 is transitive.
1 2
Thus,
The proofs that C o C2 is reflexive, symmetric, and a congruence
1
relation follow similarly and easily.
Finally, we verify that C1 o C2 is the supremum of {CIS C2}.
So,
Let
and f is one-to-one.
Let C be a congruence relation on G. Let [el denote the
equivalence class of e in C. The fact that [elc is a normal subgroup
of G is known from group theory. Let N denote [el and let f(N) = C'.
aC1b