## GR Group Theory – Factorizable Groups

Definition. A finite group $$G$$ is factored if for a positive integer $$a, b$$ With $$ab = | G |$$ There are subsets $$A, B subset G$$ cardinality $$| A | = a$$ and $$| B | = b$$ so that $$AB = G$$,

Problem 1 Is every finite group factorizable?

As I understood from these MO posts (1, 2, 3), this problem is wide open and there is no intuition as to whether it is true or not. So we can ask a relative

Problem 2 Which finite groups are factorizable?

The class of factorizable groups has a nice 3-space property:

Sentence. A finite group $$G$$ is factorizable if $$G$$ contains a normal factorizable subgroup $$H$$ with factorizable quotient $$G / H$$,

Proof. Given positive integers $$a, b$$ With $$ab = | G |$$ find whole numbers $$a_1, a_2, b_1, b_2$$ so that $$a_1a_2 = a$$, $$b_1b_2 = b$$ and $$a_1b_1 = H$$, The factorisability of $$H$$ gives two sentences $$A_1, B_1 subset H$$ cardinality $$| A_1 | = a_1$$, $$| B_1 | = b_1$$ With $$A_1B_1 = H$$, The factorisability of $$G / H$$ gives two sentences $$A_2, B_2 subset G / H$$ cardinality $$| A_2 | = a_2$$, $$| B_2 | = b_2$$ With $$A_2B_2 = G / H$$, Choose sets $$A_2, B_2 & # 39; subset G$$ so that $$| A_2 & # 39; | = | A_2 |$$, $$| B_2 & # 39; | = | B_2 |$$, $$A_2 = {aH: a in A_2 & # 39; }$$, $$B_2 = {Hb: b in B_2 & # 39;$$, Then the sets $$A = A_2 & # 39; A_1$$ and $$B = B_1B_2 & # 39;$$ Have cardinality $$| A | = a_2a_1 = a$$, $$| B | = b_1b_2 = b$$ and
$$AB = A_2? A_1B_2B_2? = A? _2HB_2 & # 39; = G$$, $$square$$

This sentence reduces the problems 1.2 to the investigation of the factorisability of finite simple groups. According to the classification of finite simple groups, each finite simple group is either cyclic or in alternative order or belongs to 16 families of Lie-type groups or is one of 26 sporadic groups.

Among these families, only the factorisability of finite cyclic groups is trivially true.

Less trivial is the following fact, which can be proved by induction.

Sentence. Every alternate group $$A_n$$ is factorizable.

Problem. Is there any hope to prove that an infinite family of simple Lie-type groups consists of factorizable groups?