(Factor Group Lemma) Let S be a generating set for a group...

(Factor Group Lemma) Let $S$ be a generating set for a group $G$, let $N$ be a cyclic normal subgroup of $G$, and let $$ \bar{S}=\{s N \mid s \in S\} $$ If $\left(a_{1} N, \ldots, a_{r} N\right)$ is a Hamiltonian circuit in $\operatorname{Cay}(\bar{S}: G / N)$ and the product $a_{1} \cdots a_{r}$ generates $N$, prove that $$ |N| *\left(a_{1}, \ldots, a_{r}\right) $$ is a Hamiltonian circuit in $\operatorname{Cay}(S: G)$.

(Factor Group Lemma) Let $S$ be a generating set for a group $G$, let $N$ be a cyclic normal subgroup of $G$, and let
$$
\bar{S}=\{s N \mid s \in S\}
$$
If $\left(a_{1} N, \ldots, a_{r} N\right)$ is a Hamiltonian circuit in $\operatorname{Cay}(\bar{S}: G / N)$ and the product $a_{1} \cdots a_{r}$ generates $N$, prove that
$$
|N| *\left(a_{1}, \ldots, a_{r}\right)
$$
is a Hamiltonian circuit in $\operatorname{Cay}(S: G)$.

Key Concepts

Factor Group Lemma

The Factor Group Lemma is a result that allows one to lift certain properties or structures from a quotient group back to the original group. In particular, if a Hamiltonian circuit exists in the Cayley graph of the quotient group and certain conditions regarding the generating set and the normal subgroup are met, then a corresponding Hamiltonian circuit can be constructed in the Cayley graph of the original group.

Quotient Group

A quotient group is formed by partitioning a group into disjoint subsets based on a normal subgroup. This concept allows one to factor out complexities of the original group and study a simplified version of the group, which is crucial in arguments where properties such as Hamiltonian circuits can be lifted from the quotient group to the entire group.

Cyclic Normal Subgroup

A cyclic normal subgroup is a special type of normal subgroup that is generated by a single element. Its cyclic nature ensures that the subgroup has a particularly simple structure, which can often be exploited when working with quotient groups or lifting properties from the quotient group back to the original group.

Normal Subgroup

A normal subgroup of a group is a subgroup that is invariant under conjugation by any element of the entire group. This property is fundamental in group theory, as it allows the construction of quotient groups, which in turn helps simplify problems by reducing the group structure via equivalence classes.

Hamiltonian Circuit

A Hamiltonian circuit in a graph is a cycle that visits every vertex exactly once before returning to the starting vertex. In the context of Cayley graphs, finding such circuits is related to exploring the group’s structure through its generating elements, and their existence can indicate deeper properties about the group and its generating set.

Cayley Graph

A Cayley graph is a representative graphical depiction of a group where each vertex corresponds to a group element and edges connect elements that differ by multiplication with a generator from a chosen generating set. This visualization helps in understanding the structure and properties of groups and can be used to explore concepts such as connectivity, symmetry, and the existence of cycles like Hamiltonian circuits.

Generating Set

A generating set of a group is a subset of group elements such that every element of the group can be expressed as a product (or combination) of these elements and their inverses. It plays a crucial role in both abstract group theory and in constructing Cayley graphs, as the edges of the graph are defined based on the elements of the generating set.

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