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properties of conjugacy
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Let $S$ be a nonempty subset of a group $G$ . When $g$ is an element of $G$ , a conjugate of $S$ is the subset $$gSg^{-1} \;=\; \{gsg^{-1}\,\vdots\;\; s \in S\}.$$ We denote here
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(1) |
If $T$ is another nonempty subset and $h$ another element of $G$ , then it's easily verified the formulae
- $(ST)^g \;=\; S^gT^g$
- $(S^g)^h \;=\; S^{gh}$
The conjugates $H^g$ of a subgroup $H$ of $G$ are subgroups of $G$ , since any mapping $$x \mapsto gxg^{-1}$$ is an automorphism (an inner automorphism) of $G$ and the homomorphic image of group is always a group.
The notation (1) can be extended to
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(2) |
where the angle parentheses express a generated subgroup. $S^G$ is the least normal subgroup of $G$ containing the subset $S$ , and it is called the normal closure of $S$ .
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"properties of conjugacy" is owned by pahio.
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Cross-references: normal subgroup, angle, homomorphic image of group, inner automorphism, automorphism, mapping, subgroup, conjugate, group, subset
There are 3 references to this entry.
This is version 2 of properties of conjugacy, born on 2009-05-25, modified 2009-05-25.
Object id is 11799, canonical name is PropertiesOfConjugacy.
Accessed 364 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
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Pending Errata and Addenda
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