Is group lasso equivalent to ridge regression when there is 1 group On Wikipedia, it says that: "while if there is only a single group, it reduces to ridge regression" (https://en.wikipedia.org/wiki/Lasso_(statistics)#Group_lasso).
However in group lasso we have norm 2, but in ridge regression we have norm 2 squared. Thanks to norm 2 squared we can obtain closed-form solution for ridge regression. I do not think we can do the same for group lasso with just 1 group.
Am I missing something or the 2 things are actually not equivalent? I am interested in solving group lasso with 1 group efficiently (as part of a bigger problem). 
 A: To see that group LASSO gives a similar solution to ridge regression upto a square in the penalty function, you need to look at subgradient conditions that characterize the solution of the group LASSO estimator. The best reference for this purpose I think is http://statweb.stanford.edu/~tibs/ftp/sparse-grlasso.pdf
These conditions are equation (4) and (5) in the paper, which are
$$ \|X^{\top}_l (y - \sum_{k\neq l} X_k \hat\beta_k)\| < \lambda \qquad \text{  (1)} \qquad \text{gives you condition when } \hat\beta_l=0 $$
and
$$ \hat\beta_l = \left(X^{\top}_l X_l - \frac{\lambda}{\|\hat\beta_l \|} \right)^{-1} X^{\top}_l r \qquad (2) \qquad \text{gives solution if } \hat\beta_l\neq0$$
where
$$ r = y - \sum_{k\neq l} X_k \hat\beta_k$$
Here l stands for a group index and X can be partitioned to non-overlapping groups. If you have a single group, these conditions boil down to
$$ \|X^{\top} (y - X \hat\beta)\| < \lambda \qquad \text{when } \hat\beta=0$$
and
$$ \hat\beta = \left(X^{\top} X - \frac{\lambda}{\|\hat\beta \|} \right)^{-1} X^{\top} y \qquad \text{otherwise }$$
We can now clearly see why group LASSO with a single group is, in fact, ridge regression with the weighted penalty term. The easiest way to solve group LASSO with a single group would be to use efficient implementations of group LASSO in whatever software you use setting the group index accordingly. Else, you could iterate the last equation a few times until the coefficient vector does not change too much. Hope this helps.
