A subgroup H of a group G is a retract of G if there exists a homomorphism [itex]q\,:\,G \rightarrow H[/itex] such that [itex]q(h) = h[/itex] for all [itex]h \in H[/itex]. This map, q, is called the retraction from G onto H. If my definition of a retract is correct then could I form a subgroup, K, of G that consists of the kernel of the retraction? That is, [tex]\mbox{ker}(q) = K < G[/tex] So K is the subgroup consisting of all elements in G that get mapped to the identity of H. Obviously, this is a normal subgroup of G and we have [itex]G = KH[/itex] and [itex]K \cap H = \{e_H\}[/itex] (from wikipedia). The group G, then, should be the semi-direction product of K and H. Is this right? Now, since H acts on K by conjugation: [tex]k \mapsto hkh^{-1}[/tex] this defines a group homomorphism [tex]p\,:\,H \rightarrow \mbox{Aut}(K)[/tex] In other words, given a group G, and a subgroup H, one can set K to be the subgroup of G consisting of elements of G that get mapped to the identity of H under the retraction. This set K is then normal, and one then has a homomorphism, p, from H into the automorphism group of K. Then the semi-direct product [itex]K \rtimes [/itex] is a group consisting of pairs [itex]hk[/itex] with multiplication [tex](h_1k_1)\cdot (h_2k_2) = (h_1 h_2)(p(h_1)(k_1)k_2)[/tex] and we also get [tex]hkh^{-1} = p(h)(k)[/tex] Now, my main question (and the reason why I brought the semi-direct product up) is this: Is the existence of a group G, which is not simple (that is, a group whose normal subgroups are not necessarily the trivial group and the group itself) and whose only only retracts are G itself, and the trivial subgroup, possible? I figured that such a group did exist. It had to be the following things: 1] It has to be a group. 2] It must not be simple. 3] It must have trivial retracts. I figured that the semi-direct product [itex]K \rtimes H[/itex] was not quite what I wanted. It is a group (by definition), it is not simple because it contains K and H as subgroups (at least), but it has H as a retract! Therefore it fails #3 of the 3 restraints. Is this all correct so far? Does anyone know of a group which satisfied all three conditions? I thought the semi-direct product came pretty close.
Wow, how did you guys come up with such easy-looking answers!? Well, [itex]4\mathbb{Z}[/itex] is a subgroup of [itex]\mathbb{Z}_4[/itex] is it not? (Also, this is a normal subrgoup, no?) and [itex]4\mathbb{Z}[/itex] is certainly not a trivial subgroup. So, [itex]\mathbb{Z}_4[/itex] is non-simple because it contains normal subgroups which are not trivial. Suppose that [itex]H[/itex] is a subgroup of [itex]\mathbb{Z}_4[/itex]. H is a retract if there is a homomorphism [itex]p\,:\,\mathbb{Z}_4 \rightarrow H[/itex] such that [itex]p(h) = h[/itex] for all h in H. So do I have to check every subgroup of [itex]\mathbb{Z}_4[/itex] and see if there is a homomorphism?
Nope. No element of [itex]4\mathbb{Z}[/itex] is an element of [itex]\mathbb{Z}_4[/itex], so it can't be a subgroup. There is a canonical map [itex]\phi: \mathbb{Z} \rightarrow \mathbb{Z}_4[/itex], but [itex]\phi(4\mathbb{Z})[/itex] is a trivial subgroup of [itex]\mathbb{Z}_4[/itex].
I verified that there is indeed a homomorphism, p, from the group [itex]\mathbb{Z}_4[/itex] into the two trivial subgroups [itex]\mathbb{Z}_4[/itex] and [itex]\{e\}[/itex] satisfying the condition. How do you check other subgroups?
Of course. I dont know what I was thinking! Is [itex]\mbox{ker}(\phi)[/itex] a subgroup of [itex]\mathbb{Z}_4[/itex]? Ok, so subgroups of [itex]\mathbb{Z}_4[/itex] have to contain 0 and have to be closed under addition. {0,1,2,3} {0,2} {0} These are the only subgroups of Z_4 that I could find. Does this look right?
Well, {0,1,2,3} is a retract since there exists a homomorphism (namely the trivial homomorphism), p, such that p(0)-0, p(1)=1, p(2)=2, p(3)=3 for 0,1,2,3 in the subgroup. Also, {0} is a retract because the trivial homomorphism p(0) =0. But if [itex]\mathbb{Z}_4[/itex] then {0,2} mustn't be a retract. That means that there must not exist a homomorphism [itex]p\,:\,\mathbb{Z}_4 \rightarrow \{0,2\}[/itex] such that [itex]p(h) = h[/itex] for all h in {0,2}. But surely there is a homomorphism.
1 generates Z_4. Any homomorphism is completely determined by what it does to the generators. Where can a homomorphism Z_4 -> {your subgroup} map 1?
If it is a homomorphism, must it map 1 to 1? That is, must it map the generator of the group to the same element in the subgroup? But if the subgroup does not have 1, will the homomorphism break down?
the answer to how did i come up with my answer is: i am teaching abstarct algebra, amnd so i know the classification of abelian groups, and i know that no group of form Z/p^n is a direct sum oif any other pair of groups, as they have different numbers of elements of order p^n.
A homomorphism must map the generator somewhere, but the only choices I have are 0 and 2. I dont see anything that forces me to choose either element. Wikipedia (which says nothing about homomorphisms having anything to do with the generator) says that a homomorphism is simply a mapping which preserves the identity element. The identity element in [itex]\mathbb{Z}_4[/itex] is 0 mod 4 = {...,-8,-4,0,4,8,...} and the homomorphism h(x) = x which takes an element from [itex]\mathbb{Z}_4[/itex] to the corresponding element in {0,2} is a mapping which preserves the identity: h(0) = 0 which is in {0,2}. So this map is a homomorphism because it preserves the identity. What is wrong with this argument.
Still, to make that connection is pretty impressive. And not groups of the form [itex]\mathbb{Z}_p[/itex] but [itex]\mathbb{Z}_{p^2}[/itex]! That takes some thinking. Unfortunately I cannot see the relationship between simple groups with only trivial retracts and abelian groups which are not the direct sum of any other pair of groups. Wait..If you have an abelian group which is not the direct sum of any other pair of groups then is the main reason why you only get trivial retracts? In other words, do you get non-trivial retracts when a group is able to be expressed as a direct product of other groups?
So, there are potentially two homomorphisms from Z_4 to {0, 2}; one that sends 1 to 0, and one that sends 1 to 2. (And, in this case, both possibilities do give rise to homomorphisms)
as observed above, a group with a retract has a normal subgroup, namely the kernel of the retract, which has a "complement", n amely the image of the retract. and any such group is a semi direct product of those two subgroups. but an abelain group has only trivial semi direct products, so if an abelian group has a retract, then it is the direct product of those two subgroups, hence any abelian grouop which is not a direct product, does not have a retract. now cyclic groups are good candidates, but the chienese remainder theorem tells us that a cyclic group whose order invovkles mroe than one prime, isa prouct, e.g. Z/15 = Z/5 x Z/3. but a cyclic group of prime power order, cannot be a product, because then the order of one factor would always divide the order of the other afctor. this means that the order of the smaller factor would annihilate the group, hence it could noit anylonger be cyclic. (in a cyclic group the annihilator cannot be a proepr factor of the order of the group.)
Have you tried constructing one? Theorem: Let G and H be nontrivial groups. Then GxH has a nontrivial retract. I'm unhappy with the wording... I think what you said is wrong for nonabelian groups. But for abelian groups: Theorem: Suppose an abelian group G has a nontrivial retract H. Then G is a direct sum of nontrivial groups. (I think you should be capable of proving both of these. The first one is much easier, though)