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− | ''in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250101.png" />'' | + | {{TEX|done}}{{MSC|20}} |
| + | ''in a group $G$'' |
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− | Elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250104.png" /> for which | + | Elements $x$ and $x'$ of $G$ for which |
| + | $$ |
| + | x' = g^{-1} x g |
| + | $$ |
| + | for some $g$ in $G$. One also says that $x'$ is the result of conjugating $x$ by $g$. The power notation $x^g$ is frequently used for the conjugate of $x$ under $g$. |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250105.png" /></td> </tr></table>
| + | Let $A,B$ be two subsets of a group $G$, then $A^B$ denotes the set |
| + | $$ |
| + | \{ a^b : a \in A\,,\, b \in B \} |
| + | $$ |
| + | For some fixed $g$ in $G$ and some subset $M$ of $G$ the set $M^g = \{ m^g : m \in M\}$ is said to be conjugate to the set $M$ in $G$. In particular, two subgroups $U$ and $V$ are called conjugate subgroups if $V = U^g$ for some $g$ in $G$. If a subgroup $H$ coincides with $H^g$ for every $g \in G$ (that is, $H$ consists of all conjugates of all its elements), then $H$ is called a ''[[normal subgroup]]'' of $G$ (or an invariant subgroup, or, rarely, a self-conjugate subgroup). |
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− | for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250106.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250107.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250108.png" /> is the result of conjugating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c0250109.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501010.png" />. The power notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501011.png" /> is frequently used for the conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501012.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501013.png" />.
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− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501014.png" /> be two subsets of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501016.png" /> denotes the set
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501017.png" /></td> </tr></table>
| + | ====Comments==== |
| + | Conjugacy of elements is an [[equivalence relation]] on $G$, and the equivalence classes are the ''conjugacy classes'' of $G$. |
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− | For some fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501019.png" /> and some subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501021.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501022.png" /> is said to be conjugate to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501024.png" />. In particular, two subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501026.png" /> are called conjugate subgroups if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501027.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501029.png" />. If a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501030.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501031.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501032.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501033.png" /> consists of all conjugates of all its elements), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501034.png" /> is called a [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025010/c02501035.png" /> (or an invariant subgroup, or, rarely, a self-conjugate subgroup).
| + | The map $x \mapsto g^{-1} x g$ for given $g$ is ''conjugation by $g$'': it is an [[inner automorphism]] of $G$. |
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− | | |
− | | |
− | ====Comments====
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| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Huppert, "Endliche Gruppen" , '''1''' , Springer (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Huppert, "Endliche Gruppen" , '''1''' , Springer (1967)</TD></TR> |
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)</TD></TR> |
| + | </table> |
2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]
in a group $G$
Elements $x$ and $x'$ of $G$ for which
$$
x' = g^{-1} x g
$$
for some $g$ in $G$. One also says that $x'$ is the result of conjugating $x$ by $g$. The power notation $x^g$ is frequently used for the conjugate of $x$ under $g$.
Let $A,B$ be two subsets of a group $G$, then $A^B$ denotes the set
$$
\{ a^b : a \in A\,,\, b \in B \}
$$
For some fixed $g$ in $G$ and some subset $M$ of $G$ the set $M^g = \{ m^g : m \in M\}$ is said to be conjugate to the set $M$ in $G$. In particular, two subgroups $U$ and $V$ are called conjugate subgroups if $V = U^g$ for some $g$ in $G$. If a subgroup $H$ coincides with $H^g$ for every $g \in G$ (that is, $H$ consists of all conjugates of all its elements), then $H$ is called a normal subgroup of $G$ (or an invariant subgroup, or, rarely, a self-conjugate subgroup).
Conjugacy of elements is an equivalence relation on $G$, and the equivalence classes are the conjugacy classes of $G$.
The map $x \mapsto g^{-1} x g$ for given $g$ is conjugation by $g$: it is an inner automorphism of $G$.
References
[a1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |
[a2] | D. Gorenstein, "Finite groups" , Chelsea, reprint (1980) |