I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|}
\hline
\circ&amp; a &amp; b &amp; e \\ \hline
 a&amp; b&amp;e &amp;a\\ \hline
 b&amp; e &amp;a &amp;b\\ \hline
 e&amp;  a&amp; b&amp;e\\ \hline
\end{array} \end{align}

Now $(G, \circ)$ forms a group. When talking about homomorphisms I learned that a mapping is considered a homomorphism if $\varphi: G \to G'$ is a mapping such that $\forall a,b \in G, \varphi(a \circ b) = \varphi(a) \circ' \varphi(b)$.

So I was asked to show that the group $(G, \circ)$ of order 3 is isomorph to the group $(G', \circ ')$ where $G' = \lbrace c,d,e' \rbrace$. So what I did was define the mapping \begin{align} \varphi: \begin{cases} &amp;G \longrightarrow G' \\ &amp;a \longmapsto c \\&amp;b \longmapsto d \\ &amp;e \longmapsto e' \end{cases} \end{align}

So it is trivial to notice that $\varphi$ is bijective. It also follows easily (I've done the calculations) that $\varphi$ is homomorphism, using the same Cayley table as above and replacing all letters by $c,d,e'$ respectively. Therefore $\varphi$ is an isomorphism. 



My question now is, what have I done? If this approach was even remotely correct, then I wonder what my statement now is saying. I have found an isomorphism between $G$ and $G'$ and therefore they have the same structure. But I defined them from the beginning as groups of the same order and even used the same Cayley-table. 

The only relevance I can see in this is that I made no statement about $c,d,e'$ at all, so I never said that they must be equal or even related to $a,b,e$.

To really hit the point home about my confusion and the triviality I see in what I have done let me get a bit more verbal. I consider the Cayley table above as regular grid, or as some sort of game in which for whatever reasons the field are labeled just as above. Now I meet my friend and replace the abstract letters $a,b,e$ by $c,d,e'$ and tell him, "look, game 1 and game 2 have the same structure!"... well, yes? 

Also my tutor said that this doesn't work for groups of order 4, can someone maybe tell me why or link me something where I can read into that?


Question: dwebp increasing the jpeg original jpeg file size
Answer: Transcoding between lossy formats tends to increase the size unless the representation of data happens to be extremely compatible between the formats, be it audio, pictures, video or other lossy data. WebP uses a 4x4 Hadamard transform, whereas JPEG uses an 8x8 Discrete Cosine Transform (DCT). Quantization, which is the main form of data loss in these formats, produces different kind of artefacts in these transformations, and transcoding cannot be optimal. Particularly, if either WebP or JPEG was saved with extremely low quality, the other format will struggle to compete with it after transcoding -- the later format will not only have to codify the image signal, but the resulting artefacts from the other format, too.

So, while there is an inherent tendency for an increase in file size in such back-and-forth conversion, the exact amount of loss happening at every stage can be controlled. Which flags and tools (including versions) are you using exactly?