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• Homomorphism
  * [[Homomorphism (category theory)]] - which all the following are instances of ** [[Homomorphism (group)]]
  4 KB (532 words) - 22:04, 19 October 2016
• Group
  ...he implicit operation of "[[juxtaposition]]", meaning {{M|ab}} denotes the group's operation applied to the elements {{M|a\in G}} and {{M|b\in G}}. !colspan="2"|For an [[Abelian group|"Abelian" or "commutative" group]]
  7 KB (1,332 words) - 07:17, 16 October 2016
• Additive function
  {{Requires references|See Halmos' measure theory book too}} ...ly additive {{M|\implies}} {{M|f}} is additive<ref name="NoStrongerClaim1" group="Note">{{Todo|Example on [[Talk:Additive function|talk page]]}}</ref>
  6 KB (971 words) - 18:16, 20 March 2016
• The fundamental group
  ...cdot)\ (\text{rel }\{0,1\})\big)} }} has a [[group]] structure, with the [[group operation]] being: ...group is actually a group|Outline of proof that {{M|\pi_1(X,b)}} admits a group structure with {{M|\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)}} a
  3 KB (393 words) - 16:10, 4 November 2016
• Paths and loops in a topological space
  * This article aims towards defining the [[Fundamental group]] ...chosen to make it distinct from paths and loops, which are terms in graph theory
  2 KB (347 words) - 19:36, 16 April 2015
• Group action
  ...ade=A|msg=Needs fleshing out and neatening up, I'd like to introduce right group actions in a different way to left, however in my current attempt they're t A (left) ''group action'' of a [[group]] {{M|(G,*)}} on a [[set]] {{M|X}} is a [[mapping]]{{rAAPAG}}:
  2 KB (320 words) - 23:28, 21 July 2016
• Homotopic maps
  ...ive to {{M|A}})''" if there exists a [[homotopy]] {{M|(\text{rel }A)}}<ref group="Note">Recall a [[homotopy]] (relative to {{M|A}}) is a [[continuous map]], *# {{M|1=\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)]}}<ref group="Note">Note that if {{M|1=A=\emptyset}} then this represents no condition/c
  4 KB (674 words) - 13:26, 15 September 2016
• Group/New page
  ...rall g\in G[g*e=e*g=g]}} - there exists an identity element of {{M|G}}<ref group="Note">At this point we do not know that the identity element is unique, th ...h*g=e]}} - for each element there exists an inverse element in {{M|G}}<ref group="Note">Again, we do not know there is a unique inverse, or for which of the
  2 KB (326 words) - 11:38, 2 July 2016
• Notes:Products and sums of groups
  ===[[Product (category theory)|Product]]=== * A [[wedge (category theory)|wedge]]: {{MM|\xymatrix{ A \\ S \ar[u]_{p_A} \ar[d]^{p_B} \\ B } }} such t
  4 KB (766 words) - 04:11, 3 July 2016
• Products and coproducts of groups
  This page will deal with computing a new group based off of 2 or more existing groups. Constructing new from old, but not {{Warning|Currently in the notes phase - see [[Notes:Products and sums of groups]]}}
  768 B (99 words) - 12:39, 7 July 2016
• Notes:Equivalence relations
  ...ng on some old abstract algebra pages, specifically involving the quotient group, and it occurs to me, a lot of this work can actually be done "higher up" t This is just some notes to get what I've done on paper into the system
  1 KB (220 words) - 16:56, 12 July 2016
• First group isomorphism theorem
  :* [[Overview of the group isomorphism theorems]] - all 3 theorems in one place ...phi) \ar@{^{(}->}[u]^i }\end{xy} }}</span></div>Where {{M|\theta}} is an [[group isomorphism|isomorphism]].
  1 KB (219 words) - 04:17, 20 July 2016
• Group factorisation theorem
  ...jection of the quotient group]], let {{M|\varphi:G\rightarrow H}} be any [[group homomorphism]], then{{rAAPAG}}: ...:G/N\rightarrow H}} given by {{M|\bar{\varphi}:[g]\mapsto\varphi(g)}} <ref group="Note">This may look strange as obviously you're thinking "what if we took
  7 KB (1,195 words) - 22:55, 3 December 2016
• Notes:Proof of the first group isomorphism theorem
  ...and {{M|H}} be [[group|groups]], let {{M|\varphi:G\rightarrow H}} be any [[group homomorphism]], then: * There exists a [[group isomorphism]], {{M|\theta:G/\text{Ker}(\varphi)\rightarrow\text{Im}(\varphi
  3 KB (528 words) - 17:41, 16 July 2016
• Free monoid generated by
  ...(especially in the case of the [[free semigroup generated by]] {{M|X}}<ref group="Note">We do this because the semigroup has no identity (in fact, is consid ** {{Warning|The "word" terminology may be specific to the [[free group]], however I wouldn't be surprised if word is used in this context too, so
  2 KB (419 words) - 16:20, 20 July 2016
• Exercises:Mond - Topology - 1/Question 8
  Which theorem of [[Group Theory (subject)|group theory]] does this resemble? ...ism (a homeomorphism is the term for a topological [[isomorphism (category theory)|isomorphism]]. Here are diagrams:
  3 KB (413 words) - 00:13, 12 October 2016
• Concatenation of paths and loops (homotopy)
  ...q(0)}} - the terminal point of {{M|p}} is the initial point of {{M|q}}<ref group="Note">Or, if they're both loops, we could just say "both loops have the sa Don't be over-eager and think "I see the [[group]] structure!" the [[constant loop]] is the identity and for a path {{M|p}}
  2 KB (351 words) - 00:57, 15 October 2016
• Ring/New page
  ...elements {{M|0_R\in R}} and {{M|1_R\in R}} (not necessarily distinct)<ref group="Note">So we could have {{M|1=0_R=1_R}} or we could have {{M|1=0_R\ne 1_R}} * {{M|(R,\oplus,0_R)}} is an [[abelian group]]
  4 KB (728 words) - 16:29, 19 October 2016
• Module
  Let {{M|(R,+,*,0)}}<ref group="Note">Or {{M|(R,+,*,0,1)}} if the ring has unity. Standard notation</ref> * An [[Abelian group]], {{M|(M,\oplus)}} together with a
  1 KB (246 words) - 22:40, 19 October 2016
• Direct product module
  ...}M_\alpha}} for convenience). This is a standard [[Cartesian product]]<ref group="Note">An alternate construction is that {{M|\prod_{\alpha\in I}M_\alpha}} *# ''Addition: ''<ref group="Note">the operation of the [[Abelian group]] that makes up a module</ref> {{M|+:M\times M\rightarrow M}} by {{M|+:((x_
  3 KB (431 words) - 22:19, 19 October 2016