Question: I cannot prove by my self the following remark

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> By the adjoint functor theorem for posets, having either all joins or all meets is sufficient for the other [1]
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So far I got this:

Let $A$ be a poset and suppose it has all lubs, for every $X \subseteq A$ let $LX$ be the lowest upper bound. When $A$ is considered as a category, $L$ is a functor from the powerset of $A$ to $A$ (morphisms are mapped to the univeral map). Since every triangle commutes in a poset, (co)limits of (co)cones are just (lubs) glbs thus $A$ is co-complete. Also $L(\{\})$ is initial. The functor $L$ preserves colimits so it has a right adjoint $G$. Thus we have adjunction, for all $X \in \mathcal P A$, $a \in A$,

$$\frac{X \le G a}{LX \le a}$$

taking the dual of this (and using the fact the opposite in powerset is just compliment) gives

$$\frac{a^{op} \le LX^{op}}{(G a)^c \le X^c}.$$

Can I conclude from this that $A^{op}$ has colimits (lubs)? I don't know how.

Otherwise I thought of using this lemma $A$ has limits of shape $J$ iff the constant functor $\Delta : A \to [J,A]$ has a right adjoint, but I don't know how to do this argument either.

[1] <http://ncatlab.org/nlab/show/complete+lattice>

Answer: 
In this particular case you could argue like this to avoid computations. By Sylow's theorem $A\_4$ has a subgroup of order $4$. It cannot be cyclic because there is no element of order $4$ in $A\_4$. The only noncyclic group of order $4$ is $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, so the statement follows.
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Is there any intuitive vision on groups,
either general or in this particular case,
that I'm missing and could help me "see" these things?

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Groups measure symmetry. We can understand groups by their groups actions, which are symmetries of a set (usually with additional structure).

Informally speaking, the set of all invertible maps from some mathematical object to itself which preserve some property will form a group. In fact, every group can be expressed in this way (via a rather formal-looking definition using category theory). In other words, every group is the automorphism group of some mathematical object.

This description of groups might not be obvious for every group you have seen so far, but do you see how symmetric groups fit this explanation? Dihedral groups? Matrix groups? What sets are they the symmetries of? What type of structure do we require these symmetries to preserve?