I know the basic group theory definition of the image of a subset of group elements under a homomorphism is the following:
The image of any subset $X \subseteq G$ is given by $\phi(X) = \{\phi(x) : x \in X\}$, and we denote $\phi(G)$ by $\text{im}\phi$.
I'm currently interested in modifying this definition to consider the "multiset image", because I want to maintain the cardinality of the set even once it's transformed under homorphism. So, for example, if we have the set $X = \{0, 1, 5\}$ in $(\mathbb Z_8, +)$, and the homomorphism $f(x) = x \;\text{mod}\; 2$, then the image by the above definition would be $\{0, 1\}$ because both 1 and 5 map to 1 so we get some amount of "collapse". I'd like to have a different definition where the image is just: $\{0, 1, 1\}$, something like:
The multiset image of any subset $X \subseteq G$ is given by the multiset $[\phi(x) : x \in X]$
so that we can allow the duplicate elements. My question is whether this notion has a name and has been studied in group theory literature. I've tried searching for "multiset image" + "homomorphism" and struggling to turn up anything, but I'd be quite surprised if this isn't a thing?
Any links to papers or better definitions/terminology than mine would be much appreciated.
Thanks!
A possible way to proceed would be to identify your subset $X$ of $G$ with the element $\sum_{g \in X} g$ of the group algebra ${\Bbb Z}[G]$. The elements of $\Bbb Z[G]$ are written as formal sums $\sum_{g \in G} c_g g$, with $c_g \in \Bbb Z$ for all $g \in G$. Now a group morphism $\varphi: G \to H$ induces a $\Bbb Z$-algebra morphism and then $\varphi(X)$ would be $\sum_{x \in X}\varphi(x)$.
In your case, you would have $G = {\Bbb Z/8\Bbb Z}$, $H = {\Bbb Z/2\Bbb Z}$ and $\varphi: G \to H$ defined by $\varphi(1) = 1$. Underlying the elements of $G$ and $H$ to avoid confusion, $X$ would be $\underline{0} + \underline{1} + \underline{5}$ and $\varphi(X)$ would be $\underline{0} + \underline{1} + \underline{1} = \underline{0} + 2\cdot \underline{1}$.