nanoSoufflé network *du Plan 9*

Is every group a subgroup of some group other than itself?

The question is dependent upon what we mean by "is": it is clear that isomorphism is an equivalence relation, so we might be so permissive as to say that a group G "is" another group H, if G ≈ H.

**In the finite case:**

Since any group G is isomorphic to some subgroup of G² (the Cartesian
product of G with itself), there is a group other than G of which it is
a subgroup.

Hahn-Banach and its consequences.

Let X be a real vector space and ρ:X→ℝ a sublinear functional such that ρ(αx + (1-α)y) ≤ αρ(x) + (1-α)ρ(y) for all x,y in X and α in [0,1]. Suppose that there is a linear functional λ:U→ℝ, for some subspace U of X such that λ(u) ≤ ρ(u) for all u in U. Then, there exists a linear functional Λ:X→ℝ such that Λ(u) = λ(u) for all u in U and Λ(x) ≤ ρ(x) for all x in X.

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