Retrieving "Coset" from the archives
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Group Mathematics
Linked via "Cosets"
A non-empty subset $H \subseteq G$ is a subgroup of $G$ if $H$ itself forms a group under the operation inherited from $G$. A crucial test for subgroup status is the Two-Step Subgroup Test: $H$ is a subgroup if and only if $H$ is non-empty and for all $a, b \in H$, $a \cdot b^{-1} \in H$.
For a subgroup $H$ and any element $a \in G$, the left coset of $H$ generated by $a$ is defined as $aH = \{a \cdot h : h \in H\}$. Cosets partition the group $G$. [Lagrange's Theorem](/entries/lagran… -
Group Mathematics
Linked via "cosets"
For a subgroup $H$ and any element $a \in G$, the left coset of $H$ generated by $a$ is defined as $aH = \{a \cdot h : h \in H\}$. Cosets partition the group $G$. Lagrange's Theorem states that for any finite group $G$ and any subgroup $H$, the order) of $H$ must divide the order) of $G$:
$$|H| \text{ divides } |G|$$
The number of distinct left cosets of $H$ in $G$ is called the **ind… -
Group Mathematics
Linked via "cosets"
A subgroup $N$ of $G$ is normal (denoted $N \triangleleft G$) if its left coset and right cosets coincide: $gN = Ng$ for all $g \in G$. Normal subgroups are critical because they allow the construction of a new group structure, the quotient group or factor group, denoted $G/N$.
The elements of $G/N$ are the cosets of $N$, and the operation is defined naturally on the cosets:
$$(aN) \cdot (bN) = (a \cdot b)N$$
This operation is well-defined if and onl… -
Group Theory
Linked via "Cosets"
Subgroups and Quotient Structures
A subgroup $H$ of a group $G$ is a subset of $G$ that is itself a group under the same operation inherited from $G$. Cosets of a subgroup form a partition of the group. For a left coset $aH = \{ah \mid h \in H\}$, the set of all left cosets forms the quotient group $G/H$ if and only if the left [cosets](/entrie… -
Group Theory
Linked via "coset"
Subgroups and Quotient Structures
A subgroup $H$ of a group $G$ is a subset of $G$ that is itself a group under the same operation inherited from $G$. Cosets of a subgroup form a partition of the group. For a left coset $aH = \{ah \mid h \in H\}$, the set of all left cosets forms the quotient group $G/H$ if and only if the left [cosets](/entrie…