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• Kernel
  ...of the same type of space (which is imbued with an identity element), the kernel of {{M|f:X\rightarrow Y}} (where {{M|f}} is a [[Function|function]]) is def ...'kernel'' where {{M|Y}} is a space imbued with the concept of identity<ref group="footnotes">Ambiguous for [[Field|fields]] as they have two identities.</re
  2 KB (376 words) - 19:53, 10 May 2015
• Normal subgroup
  Let {{M|(G,\times)}} be a [[Group|group]] and {{M|H}} a [[Subgroup|subgroup]] of {{M|G}}, we say {{M|H}} is a '''no ...the kerel of some [[Homomorphism|homomorphism]] of {{M|G}} into some other group
  5 KB (1,026 words) - 18:07, 17 May 2015
• Notes:Group isomorphism theorems
  ...homomorphism]] (so {{M|H}} is also a group), whose [[kernel (group theory)|kernel]] contains {{M|N}}, then {{M|\varphi}} [[factor (function)|factors]] unique That is to say that there exists a group homomorphism:
  4 KB (654 words) - 23:06, 10 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
  ...n 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
• An injective group homomorphism means the group is isomorphic to its image
  : This is a [[corollary]] to the [[first group isomorphism theorem]] Suppose {{M|\varphi:A\rightarrow B}} is any ''[[injective]]'' [[group homomorphism]], then{{rAAPAG}}:
  4 KB (727 words) - 04:53, 20 July 2016
• Notes:Homology
  # '''{{M|\mathbf{n} }}^th homology group: ''' {{M|1=H_n:=Z_n/B_n}} * We extend this to a [[group homomorphism]] by defining:
  6 KB (897 words) - 07:30, 15 October 2016
• Exercises:Mond - Topology - 1/Question 8
  Which theorem of [[Group Theory (subject)|group theory]] does this resemble? ...eory (subject)|categorical]] sense. The group theorem "factors through the kernel of the morphism" where as this "factors through the equivalence relation in
  3 KB (413 words) - 00:13, 12 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
• Quotient module
  * The [[quotient group]] {{M|\frac{M}{A} }} is actually a (left) [[module]] too with the operation *# (ADDITION) - given by the quotient group part
  1 KB (209 words) - 20:00, 23 October 2016
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.5
  ...group homomorphism]] given by the [[matrix]] on the right. Calculate its [[kernel]], [[image]] and [[cokernel]]. ** {{XXX|Cokernel is the homology group here. Check that}}
  3 KB (473 words) - 19:34, 14 January 2017
• Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
  ...n^\Delta(T^2)\cong 0}}<ref group="Note">Here {{M|0}} denotes the [[trivial group]]</ref> for {{M|n\in\{3,\ldots\}\subset\mathbb{N} }} ...the generators (which are extended to group homomorphisms on free Abelian group by the characteristic property of free Abelian groups):
  10 KB (1,664 words) - 12:43, 1 March 2017