1
vote
0answers
45 views

ZigZag product - A simpler definition?

I have been fiddling with the ZigZag product and constructing expanders for a while now. I was wondering if the following definition of a ZigZag product is the same as the original article: Lets ...
2
votes
0answers
30 views

Is there a sort of “two-sided semidirect product”? [duplicate]

Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter. What can we ...
3
votes
1answer
65 views

Direct product commutes with direct sum?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
2
votes
0answers
80 views

Multiplicative group into ring operation

My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). ...
2
votes
1answer
102 views

To What Extent Does the Cartesian Product for Algebraic Structures Generalize?

I admit this question is quite general. If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
0
votes
1answer
129 views

Weak direct product

I am just reading the book "Algebra" by Hungerford and on one page it says that if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$ where ...
3
votes
2answers
117 views

non-homomorphic algebras

On the page http://planetmath.org/encyclopedia/DirectProduct9.html I have read, that we may embed $\boldsymbol{A}$ into $\boldsymbol{A}\times \boldsymbol{B}$ if there exists homomorphism ...
3
votes
1answer
323 views

Prove that the dihedral group $D_4$ can not be written as a direct product of two groups

I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, ...