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I know this

every group is a normal subgroup of itself
every group has identity element
every normal subgroup has identity element

The kernel of a group homomorphism $f:G\rightarrow G^{'}$ is the set of all elements of $G$ which are mapped to the identity element of $G^{'}$. The kernel is a normal subgroup of $G$, and always contains the identity element of $G$. It is reduced to the identity element iff f is injective.

Identity element is called also unit element or 1

What kind of structure do I have if I don’t have the identity element (unit element or 1) ?

I know that a group without identity element is a semigroup. My problem is to classify normal subgroup when I 'delete' identity element.

The normal subgroup in what does it become if I now have a semigroup?

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Jack
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  • What exactly is a normal subgroup without identity? If $N\trianglelefteq G$, that means that for $g\in G$, $gNg^{-1}=N$. What is $g^{-1}$? I don't know what you mean. – Rushabh Mehta Sep 12 '19 at 16:00
  • You have not understood, normal subgroups must have an identity, but if I go from the group to the semigroup (that is, to a structure without identity) what is the equivalent definition of a normal subgroup in the context of a semigroup? – Jack Sep 12 '19 at 16:03
  • Each idempotent of the semigroup acts like an identity on a subset of the semigroup. They are like stars and the subgroups of the semigroup, which might not share an idempotent, are like solar systems. – Shaun Sep 12 '19 at 16:03
  • @Shaun Do you mean that equivalent of 'normal subgroups' of a group is, in context of semigroups, an idempotent semigroup ?https://www.jstor.org/stable/pdf/2307797.pdf?seq=1#page_scan_tab_contents – Jack Sep 12 '19 at 16:06
  • No, @Jack. That's not what I mean. I don't know what the analogue of a normal subgroup is in terms of semigroups, only subgroups. – Shaun Sep 12 '19 at 16:07
  • @Jack No I get that, but I don't even think there can be an equivalent, since the notion is so dependent on the existence of an identity. – Rushabh Mehta Sep 12 '19 at 16:09
  • Ok, but if I 'downgrade' normal subgroup "deleting" identity element in a group, what happens for normal subgroup ? – Jack Sep 12 '19 at 16:10
  • @Jack Nothing in particular. Why would you expect that to produce something meaningful> – Rushabh Mehta Sep 12 '19 at 21:08

1 Answers1

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A semigroup is just a group with no requirement of having an identity, or inverses. Simply "deleting" the identity in a group will not in general produce a semigroup. For example, the set $\mathbb{Z}\setminus\{0\}$ is not closed under addition.

79037662
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  • I try to find the analogue of a normal subgroup is in terms of semigroups. Exists ? – Jack Sep 12 '19 at 16:09
  • if I 'downgrade' normal subgroup "deleting" identity element in a group, what happens for normal subgroup ? – Jack Sep 12 '19 at 16:11
  • @Jack Well the definition of normal subgroup relies on the concept of inverses, so I'm gonna say no. – 79037662 Sep 12 '19 at 16:11
  • @Jack "Deleting" the identity element will make the set no longer a group (or even a semigroup, as I said in my answer). Therefore the concept of "subgroup", let alone "normal subgroup", becomes meaningless. – 79037662 Sep 12 '19 at 16:13
  • Ok, it becomes meaningless but in what structure this meaningless "normal subgroup" it becomes ? I still get something, even if in something completely different the previous concept of "normal subgroup" ? – Jack Sep 12 '19 at 16:15
  • That's like asking "what is the colour of 5". If you just delete the identity element of a group, what's left will in general have no analogue of "normal subgroup" because the whole concept of "normal subgroup" relies on the existence of inverses and identity. – 79037662 Sep 12 '19 at 16:19
  • ah..ok, you mean normal subgroup is defined both inverse and identity element. You say that if I remove identity I also remove inverse element meaning that give a meaning to exist the concept of "normal subgroup". In other words, I remove inverse and identity element, not only identity element, is right ? – Jack Sep 12 '19 at 16:27
  • So, if I remove the whole concept of normal subgroup from a group what kind of structure I get ? – Jack Sep 12 '19 at 16:27
  • I don't understand what you're asking. – 79037662 Sep 12 '19 at 16:33
  • I have this definition every group is a normal subgroup of itself. I change definition in this way removing 'normal subgroup' every group is a ..... .......... of itself. What happens ? What structure I get now ? I remove inverse element and identity element because as you say whole concept of "normal subgroup" relies on the existence of inverses and identity – Jack Sep 12 '19 at 17:01
  • I remove only inverse and identity element from group definition (that is like to remove normal subgroup from group definition), not other operations. What structure I get ? – Jack Sep 12 '19 at 17:05
  • Take a look here https://math.stackexchange.com/questions/1053403/example-of-an-associative-binary-operation-without-identities-or-inverses . in my situation I need to downgrade from a group, in last answer exist a situation for rings without identity and without inverses, but in my situation I start from group definition – Jack Sep 12 '19 at 17:09
  • I suppose you'd get a semigroup, if you lose the conditions of identity and inverses. A semigroup is a subsemigroup of itself I suppose, but that doesn't really have anything to do with "normality". – 79037662 Sep 12 '19 at 17:45
  • Maybe should be is a partial semigroup (closure maybe is missed) – Jack Sep 12 '19 at 23:52