FoldUnfold Table of Contents The Internal Direct Product of Two Groups The Internal Direct Product of Two Groups Recall from The External Direct Product of Two Groups page that if $(G, *)$ and $(H, +)$ are two groups then the external direct product of these groups is the new group $(G \times H, \cdot)$ where […]

## The Canonical Injections of Weak Direct Products of Groups

FoldUnfold Table of Contents The Canonical Injections of Weak Direct Products of Groups The Canonical Injections of Weak Direct Products of Groups Recall from The Weak Direct Product of an Arbitrary Collection of Groups page that if $\{ G_i : i \in I \}$ is an arbitrary collection of groups the the weak direct product […]

## The Weak Direct Product of an Arbitrary Collection of Groups

FoldUnfold Table of Contents The Weak Direct Product of an Arbitrary Collection of Groups The Weak Direct Product of an Arbitrary Collection of Groups Definition: Let $\{ G_i : i \in I \}$ be an arbitrary collection of products. Define $\displaystyle{\prod_{i \in I}^{\mathrm{weak}} G_i = \left \{ f : I \to \bigcup_{i \in I} G_i […]

## The Canonical Projections of Direct Products of Groups

FoldUnfold Table of Contents The Canonical Projections of Direct Products of Groups The Canonical Projections of Direct Products of Groups Recall from The Direct Product of an Arbitrary Collection of Groups page that if $\{ G_i : i \in I \}$ is an arbitrary collection of groups then the direct product of these groups is […]

## The Direct Product of an Arbitrary Collection of Groups

FoldUnfold Table of Contents The Direct Product of an Arbitrary Collection of Groups The Direct Product of an Arbitrary Collection of Groups Recall from The Direct Product of Two Groups page that if $(G, \cdot)$ and $(H, *)$ are groups then the direct product of these groups is another group, $G \times H$ with the […]

## The Order of an Element in an External Direct Product of Two Groups

FoldUnfold Table of Contents The Order of an Element in an External Direct Product of Two Groups The Order of an Element in an External Direct Product of Two Groups Recall from The External Direct Product of Two Groups page that if $(G, *)$ and $(H, +)$ are two groups then the external direct product […]

## The Direct Product of Two Groups

FoldUnfold Table of Contents The Direct Product of Two Groups The Direct Product of Two Groups Suppose that $(G, \cdot)$ and $(H, *)$ are two groups. We would like to obtain a new group on the cartesian product $G \times H = \{ (g, h) : g \in G, h \in H \}$. There is […]

## The Order of HK of Two Subgroups H and K of a Group G

FoldUnfold Table of Contents The Order of HK of Two Subgroups H and K of a Group G The Order of HK of Two Subgroups H and K of a Group G Recall from The Product HK of Two Subgroups H and K of a Group G page that if $G$ is a group and […]

## The Product HK of Two Subgroups H and K of a Group G

FoldUnfold Table of Contents The Product HK of Two Subgroups H and K of a Group G The Product HK of Two Subgroups H and K of a Group G Definition: Let $G$ be a group and let $H$ and $K$ be two subgroups of $G$. The Product of $H$ and $K$ is $HK = […]

## Criterion for a Finite Abelian Group to be Simple

FoldUnfold Table of Contents Criterion for a Finite Abelian Group to be Simple Criterion for a Finite Abelian Group to be Simple Theorem 1: Let $G$ be a finite abelian group. Then $G$ is a simple group if and only if the order of $G$ is prime. Proof: Since $G$ is an abelian group, every […]

- « Previous Page
- 1
- …
- 292
- 293
- 294
- 295
- 296
- …
- 401
- Next Page »