Pull-backs and Push-outs

"Pull-back" is a standard term in group theory.¹ "Push-out" is also used, less frequently. This page aims to explain both.

Pull-backs

Suppose we have three groups B, H and K with the following isomorphisms:
θ: B→K
φ: H→K
Consider the direct product B×H, its elements being the pairs (b,h) | b∈B, h∈H. It must have a subgroup whose elements are the pairs (b,h) | b∈B, h∈H, θ(b)=φ(h). This subgroup is isomorphic to the pull-back of B×H by K. If the subgroup is normal in B×H, then the quotient is isomorphic to K.

A simple way to build a non-trivial pull-back is to start with two groups A and G whose automorphism groups each have K as a subgroup. Then, using the notation of the previous paragraph, we have
B is A⋊K
H is G⋊K
and the pullback is (A×G)⋊K using the same K to extend both A and G.

We can regard this as . Its order is the product of the orders of A, G and K. It can be regarded in any of these ways:
    (A×G)⋊K
    A⋊(G⋊K)
    A⋊(G⋊K).

Example, order 18

The smallest non-trivial example is the pull-back of C3×C3 by C2. It has nine elements of order 2 and eight of order 3. Its direct product with C2 is isomorphic to D6×D6; each C3 has been extended by a C2 and then one of the C2s has been "pulled back".

Example, order 30

Another simple example is the pull-back of C3×C5 by C2. This is the group normally known as D30. Regarded as a pull-back, it is (D6×D10)/C2.

Push-outs

Whereas a pull-back extends two groups simultaneously, with the original groups being normal subgroups of the extensions, a push-out is the other way round. It is the central extension of two groups simultaneously, adding the same central subgroup to the centre of each.

Suppose we have two groups B and H such that C is the centre of each. Then the centre of B×H must be C×C, and we can quotient out C from B×H in a way that leaves both B and H with a centre (in fact the same centre). This quotient is the push-out of B×H by C.

More generally, suppose we have two groups B and H such that C is a subgroup of the centre of each. Then the centre of B×H must have C×C as a central subgroup, and we can again quotient out one C from B×H in a way that leaves both B and H with the same central C. This quotient is the push-out of B×H by C.

I regard the push-out of B×H by C like this. Writing B/C as A and H/C as G, the push-out is . Its order is the product of the orders of A, G, and C. It can be regarded in any of these ways:
    C↑(A×G)
    (C↑A)↑G
    (C↑G)↑A.

Formal definition of "push-out"

Thw above paragraphs use the phrases "same centre" and "same central C" in a way that I hope is clear. But in case it isn't, here is a formal definition of "push-out".

Suppose we have groups B, H and K, with K being isomorphic both to a subgroup of the centre of B and to a subgroup of the centre of K. Now the direct product B×H has as its elements the pairs (b,h) | b∈B, h∈H. This direct product has as a normal subgroup a N group isommorphic to K, defined as { (k,k^-1) | k∈K }. Then the quotient (B×H)/N is the push-out of B and H by K.

Example: Q8

The smallest example is the pull-down of (C2×C2) by C2, giving Q8. It is the quotient of C4×C4 by a central C2: if the two C4s are presented as < a | a⁴ > and < b | b⁴ >, and the central C2 that is quotiented out is {1, a²b²}, we are left with Q8 consisting of cosets as follows:

1	{1, a²b²}	-1	{a², b²}
i	{a, a³b²}	-i	{a³, ab²}
j	{b, a²b³}	-j	{a²b, b³}
k	{ab, a³b³}	-k	{a³b, ab³}

Example: the Pauli group

Another example of a pull-down is the Pauli group, which is

C2↑C2 is C4, and C2↑(C2×C2) is Q8 as we have seen above. So the group generated has
    a central C2 with C2×C2×C2 as the quotient;
    a central C4 with C2×C2 as the quotient;
    Q8 as a normal subgroup with C2 as the quotient.

Example: the Modular group of order 16

Another example of a pull-down is the order 16 modular group, which is

C2↑C2 is C4, and C2↑(C2↑C2) is C8. So the group generated has
    a central C2 with C2×C4 as the quotient;
    a central C4 with C4 as the quotient;
    C8 as a normal subgroup with C2 as the quotient.

References

1 Alejandro Adem & R. James Milgram. , Springer 2004, ISBNs 3540202838, 9783540202837. Page 19.

2 We use A↑B to denote a central extension of B by A (A being the centre), as explained at How to Build Groups

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