### July 10, 2020, 12:08 p.m.

## Web Development

### By Maurice Ticas

#### Tags:

#### Personal

Fast forward to today and we have a very large JavaScript community of web developers. I'll describe my journey into this web development community and share what I find to be the good tools to get the job done.

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### April 7, 2020, 11:52 p.m.

## Manipulating Group Elements

### By Maurice Ticas

#### Tags:

#### Algebra

#### Group Theory

#### Proof

What is there to say when given a group and a few of its elements? With very little information, group behavior of elements can reveal simple relationships. We summarize some simple relationships when manipulatiing group elements.

Consider the group \(<G,\cdot>\) and let \(a_1,a_2,a_3 \in G\). We then have that
\[a_{1}^{3} = e \Rightarrow a_1 = (a_{1}^{-1})^2\]
\[a_{1}^{2}=e \Rightarrow a_{1}=(a_{1}^{-1})^3\]
\[\text{If }(a_{1})^{-1} = a_{2}^{3} \text{ for some } a_2 \in G \text{, then } a_1 = (a_2^{-1})^3\]
\[a_{1}^{2}a_{2}a_{1} = a_{2}^{-1} \Rightarrow a_2 = \left [ (a_{1}a_{2}a_{1})^{-1} \right]^{3} \]
\[a_{1}a_{2}a_{1} = a_{3} \Rightarrow a_{2}a_{3}=(a_{2}a_{1})^{2}\]

You can read and study the proofs. They just use the properties of a group. Arguments will not be more involved than a typical usage of induction.

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### March 25, 2020, 9:32 a.m.

## Groups, Subgroups and Algebraic Structure

### By Maurice Ticas

#### Tags:

#### Algebra

#### Group Theory

Here we'll give a very high level overview of the subject to summarize the material from our Secolinsky group theory publication. Anyone wanting to better understand groups is invited to read its beginning theoretic development. The publication will help you understand the details and concrete nature of the rich subject.

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