Intereting Posts

Examples of metric spaces which are not normed linear spaces?
Fitting of Closed Curve in the Polar Coordinate.
Understanding the context free grammar of the following language
Statistics resources with examples for a C.S. student
Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$
Can an infinite set of primes be a regular language or CFG?
If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose center is $C$ be such that $CP$ is perpendicular to $CQ$
Evaluate the closed form of $\int_{0}^{\pi/2}\arctan\left(\sqrt{\sin(2x)\over \sin^2(x)}\right)\mathrm dx$
Does there exist a complex function which is differentiable at one point and nowhere else continuous?
Cardinal number subtraction
Differentiating an infinite sum
Formula for binomial coefficients
Why are the order-of-operations conventions good?
Query regarding proof of Existence Theorem (1.19, pg-17, Baby Rudin)
Combinatorial interpretation of a sum identity: $\sum_{k=1}^n(k-1)(n-k)=\binom{n}{3}$

I’m wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I’m trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. Since $\mathrm{Aut}\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\cong GL_2(\mathbb{F}_p)$, I want to specify $\varphi:S_3\rightarrow\mathrm{Aut}\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)$ as $\varphi:S_3\rightarrow GL_2(\mathbb{F}_p)$ element by element (i.e. give a 2×2 matrix for each element of $S_3$).

I want to do this just to do group element multiplications quickly. I’ll obviously specify $p$ when necessary.

I’m fairly new to GAP so a low-level answer (or reference) would be appreciated.

- Can you recommend a decent online or software calculator?
- What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?
- Construction of formula in Sagemath program
- Computing Invariant Subspaces of Matrix Groups
- Computing 2d radially symmetric Fourier transforms (with Wolfram Alpha)
- Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

- How many groups of order $2058$ are there?
- Question concerning the GAP package qpa (find all ideals with a certain property)
- How to define a recurrence relation in Maple?
- symbolic computation program/software
- What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?
- If $h$ divides $|G|$, not necessary that $G$ has a subgroup of order $h$
- Substituting values in multivariate polynomial in GAP
- Finding all the group elements of a certain order of a finite group
- Most wanted reproducible results in computational algebra
- How getting the unitarized irreducible representations with GAP?

In case you still don’t have an answer:

You want to use the GAP command

`SemidirectProduct( <g>, <hom>, <n> )`

.

Here, `<hom>`

is a group homomorphism from the group `<g>`

to the automorphism group of the group `<n>`

.

In GAP this might look like this (for $p=7$):

```
gap> h := CyclicGroup(7);;
gap> g := DirectProduct( h, h );;
gap> aut := AutomorphismGroup( g );;
gap> iso := IsomorphismGroups( GL(2,7), aut );;
gap> s := SymmetricGroup(3);;
gap> GeneratorsOfGroup( s );
[ (1,2,3), (1,2) ]
gap> hom := GroupHomomorphismByImages( s, GL(2,7), GeneratorsOfGroup( s ), [ a, b ] );
[ (1,2,3), (1,2) ] -> [ [ [ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ], [ [ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ] ]
```

`a`

and `b`

are your favourite images of `(1,2,3)`

and `(1,2)`

, respectively. I chose `One(GL(2,7))`

because I’m lazy. Note that you have to tell GAP that your matrices are over $\mathbb{F}_p$. Fortunately, that can be done like this:

```
gap> a := [ [ 1, 2 ], [ 3, 4 ] ] * One( GF(7) );
[ [ Z(7)^0, Z(7)^2 ], [ Z(7), Z(7)^4 ] ]
gap> a in GL(2,7);
true
```

Now you can compose `hom`

and `iso`

and define your semidirect product:

```
gap> semi := SemidirectProduct( s, CompositionMapping( iso, hom ), g );
<pc group with 4 generators>
```

- Is the number of primes congruent to 1 mod 6 equal to the number of primes congruent to 5 mod 6?
- What are some interesting calculus facts your calculus teachers didn't teach you?
- Count arrays with each array elements pairwise coprime
- A family of generating functions related to generalized harmonic numbers and power functions.
- Must a Hermitian/Kähler Manifold have a complex structure?
- Zeta function values in terms of Bernoulli numbers.
- is a net stronger than a transfinite sequence for characterizing topology?
- Solving $-u''(x) = \delta(x)$
- Proving $\sum_{k=1}^n k k!=(n+1)!-1$
- What is the difference between Cartesian and Tensor product of two vector spaces
- Hamel basis for the vector space of real numbers over rational numbers and closedness of the basis under inversion
- What are the main relationships between exclusive OR / logical biconditional?
- Boolean Algebra, Simplification: Don't know the method used
- Drawing sine and cosine waves
- Holomorphic and Harmonic functions