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2013-02-15T17:31:32Z

Created page with "{{subgroup property}} {{wikilocal}} ==Definition== Suppose <math>G</math> is a group and <math>H</math> is a subgroup of <math>G</math>. We say that <math>H</math> i..."

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{{subgroup property}}
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==Definition==

Suppose <math>G</math> is a [[group]] and <math>H</math> is a [[subgroup]] of <math>G</math>. We say that <math>H</math> is '''verbally closed''' in <math>G</math> if the following is true: For any word <math>w</math> in <math>n</math> letters, the image of <math>H^n</math> under the [[word map]] corresponding to <math>w</math> equals the intersection of <math>H</math> with the image of <math>G^n</math> under the word map corresponding to <math>w</math>.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::retract]] || || || || {{intermediate notions short|verbally closed subgroup|retract}}
|-
| [[Weaker than::direct factor]] || || (via retract) || (via retract) || {{intermediate notions short|verbally closed subgroup|direct factor}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::local divisibility-closed subgroup]] || if an element of the subgroup has a <math>n^{th}</math> root in the whole group, there is a <math>n^{th}</math> root in the subgroup. || || || {{intermediate notions short|local divisibility-closed subgroup|verbal subgroup}}
|-
| [[Stronger than::divisibility-closed subgroup]] || if every element of the group has a <math>n^{th}</math> root, every element of the subgroup has a <math>n^{th}</math> root in the subgroup. || (via local divisibility-closed) || (via local divisibility-closed) || {{intermediate notions short|divisibility-closed subgroup|verbally closed subgroup}}
|-
| [[Stronger than::local powering-invariant subgroup]] || if an element of the subgroup has a unique <math>n^{th}</math> root in the group, that root is in the subgroup. || (via local divisibility-closed) || (via local divisibility-closed) || {{intermediate notions short|local powering-invariant subgroup|verbally closed subgroup}}
|-
| [[Stronger than::powering-invariant subgroup]] || || (via divisibility-closed) || (via divisibility-closed) || {{intermediate notions short|powering-invariant subgroup|verbally closed subgroup}}
|}

Verbally closed subgroup - Revision history

Vipul: Created page with "{{subgroup property}} {{wikilocal}} ==Definition== Suppose G is a group and H is a subgroup of G. We say that H i..."

Vipul: Created page with "{{subgroup property}} {{wikilocal}} ==Definition== Suppose $G$ is a group and $H$ is a subgroup of $G$ . We say that $H$ i..."