Every group is normal in itself

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This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., identity-true subgroup property)
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Statement

Statement with symbols

Suppose G is a group. Consider G as a subgroup of itself -- G is a normal subgroup of itself.

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Proof

Proof using the conjugation definition of normality

Given: A group G.

To prove: For any g \in G and h \in G, ghg^{-1} \in G.

Proof: This is obvious from the fact that G is closed under multiplication.

Proof using the kernel of homomorphism definition of normality

Given: A group G.

To prove: There exists a homomorphism \varphi:G \to K for some group K such that every element of G maps to the identity element.

Proof: Let K be the trivial group and \varphi be the map sending every element of G to the identity element of K. Clearly, \varphi satisfies the conditions for being a homomorphism: for any g,h \in G, both \varphi(gh) and \varphi(g)\varphi(h) equal the identity element of K. Moreover, every element of G is sent to the identity element of K under \varphi.

Proof using the cosets definition of normality

Given: A group G.

To prove: For every element g \in G, gG = Gg.

Proof: Note that

  1. gG = G: Clearly, gG \subseteq G. Also, for any h \in G, h = g(g^{-1}h) \in gG, so G \subseteq gG. Thus, gG = G.
  2. Gg = G: Clearly, Gg \subseteq G. Also, for any h \in G, h = (hg^{-1})g \in Gg, so G \subseteq Gg. Thus, Gg = G.

Combining the two steps, we obtain that Gg = gG.

Proof using the union of conjugacy classes definition of normality

Given: A group G.

To prove: G is a union of conjugacy classes in G.

Proof: The conjugacy classes form a partition of G (arising from the equivalence relation of being conjugate), so G is their union.

Proof using the commutator definition of normality

Given: A group G.

To prove: For every g,h \in G, [g,h] \in G.

Proof: This is direct from the fact that G is closed under multiplication and inverses, and the commutator is defined in terms of these operations.