Group (1) - Printable Version

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Group (1) - elim - 09-14-2010 01:15 PM

(1) Group $G$ is abelian if $\forall a \in G \quad (a^2 = e)$
(2) Let $(G, \cdot)$ be a semigroup. Show that
\(\quad\) (i) $G$ is a group iff $\forall a,b \in G \; \exists x,y \in G \quad(a\cdot x = b,\; y \cdot a = b)$
\(\quad\) (ii) If $|G| < \infty$, then $G$ is a group iff $(xa = xb \Rightarrow a=b) \wedge (ay=by \Rightarrow a=b)$ (cancellation laws)
(3) If $G$ is a group and $A,B \subset G,\; A\nsubseteq B,\; B\nsubseteq A$, then $A\cup B$ is not a group