Group Theory Problem

**Deveno** · October 8th 2011, 07:44 PM

this isn't all that hard, but i decided to post it here, because it's interesting.

Let P be a partition of a group G with the property that for any pair of elements A,B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P that contains the identity of G. Prove that N is a normal subgroup of G and that P is the set of its cosets.

**NonCommAlg** · October 9th 2011, 01:01 PM

Originally Posted by Deveno

this isn't all that hard, but i decided to post it here, because it's interesting.

Let P be a partition of a group G with the property that for any pair of elements A,B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P that contains the identity of G. Prove that N is a normal subgroup of G and that P is the set of its cosets.

$(*)$ for any $A,B \in P,$ there exists $C \in P$ such that $AB \subseteq C.$

note that an obvious result of $(*)$ is that for any $Q_1 \in P$ and $g, h \in G$ , there exists $Q_2 \in P$ such that $gQh \subseteq Q_2$ .

now, for any $a \in N$ we have $a = a \cdot 1 \in N^2$ and so $N \subseteq N^2$ . we also have $N^2 \subseteq Q$ for some $Q \in P$ , by $(*)$ . but then $N \subseteq Q$ and hence $N=Q$ . therefore $N=N ^2$ and so $N$ is closed under multiplication. let $a \in N$ . then $Na^{-1} \subseteq Q$ for some $Q \in P$ and since $1 \in Na^{-1}$ , we have $1 \in N \cap Q$ . thus $Q=N$ and so $Na^{-1} \subseteq N$ . therefore $a^{-1} \in N$ and so $N$ is a subgroup. now let $g \in G$ . then, by $(*)$ , there exists $Q \in P$ such that $gNg^{-1} \subseteq Q$ . but then $1 \in Q \cap N$ , because $1 \in gNg^{-1} \subseteq Q$ , and thus $Q=N$ . so we have proved that $gNg^{-1} \subseteq N$ , i.e. $N$ is normal.

finally, let $g \in G$ and choose $Q \in P$ such that $g \in Q$ . then $gN \subseteq Q_1$ for some $Q_1 \in P$ , by $(*)$ . but then $g \in Q_1 \cap Q$ , because $g \in Ng \subseteq Q_1$ , and so $Q_1=Q$ . hence

$g N \subseteq Q.$

let $Q_2 \in P$ be such that $N \subseteq g^{-1}Q \subseteq Q_2$ . then $1 \in Q_2 \cap N$ , because $1 \in g^{-1}Q \subseteq Q_2$ , and so $Q_2=N$ . hence $gN=Q$ and we are done.

**Deveno** · October 9th 2011, 01:29 PM

G is not given to be finite, so $N = N^2$ is insufficient to establish that N is a subgroup.

**NonCommAlg** · October 9th 2011, 02:19 PM

Originally Posted by Deveno

G is not given to be finite, so $N = N^2$ is insufficient to establish that N is a subgroup.

ok, you're right but that can be easily fixed. see the solution again.
the problem is nice but the idea of the solution is very simple used over and over again!

**Deveno** · October 9th 2011, 02:44 PM

what i found "interesting" about it, was not so much its difficulty level, but the idea that if a partition respects the group multiplication, not only must it be a co-refinement (is that even a word?) of the coset partition, but in fact they are equal.

i'm pretty sure that this is equivalent to saying that every congruence is in fact a coset partition.

and yeah, once you establish N is one element of the partition, g-translation pretty much wraps up the rest.

**ymar** · October 22nd 2011, 12:16 PM

Originally Posted by Deveno

i'm pretty sure that this is equivalent to saying that every congruence is in fact a coset partition.

Indeed, if I'm not mistaken, the problem is a restatement of the well-known theorem stating the relation between congruences on groups and normal subgroups.

One of the two common definitions of a congruence on a semigroup is that it's an equivalence relation R such that, if (a R b) and (c R d), then (ac R bd). The hypothesis of the problem is clearly (I think but I'm often wrong) equivalent to the statement that P is a partition given by a congruence.

Thread: Group Theory Problem

LinkBack

Thread Tools

Display

Group Theory Problem

Re: Group Theory Problem

Re: Group Theory Problem

Re: Group Theory Problem

Re: Group Theory Problem

Re: Group Theory Problem

Similar Math Help Forum Discussions

challenging group theory problem

Group theory problem with congruence relations

Tough Problem from Rotman's Group Theory, 3rd Ed

Group theory problem

Another Group Theory problem...

Search Tags