diff --git a/mysql-test/r/group_min_max.result b/mysql-test/r/group_min_max.result
index a43f1ee88a6..13eb2690e86 100644
--- a/mysql-test/r/group_min_max.result
+++ b/mysql-test/r/group_min_max.result
@@ -1954,6 +1954,24 @@ id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
 explain select sum(ord(a1)) from t1 where (a1 > 'a') group by a1,a2,b;
 id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
 1	SIMPLE	t1	index	idx_t1_0,idx_t1_1,idx_t1_2	idx_t1_1	163	NULL	128	Using where; Using index
+explain select a1 from t1 where a2 = 'b' group by a1;
+id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
+1	SIMPLE	t1	range	NULL	idx_t1_1	130	NULL	5	Using where; Using index for group-by
+select a1 from t1 where a2 = 'b' group by a1;
+a1
+a
+b
+c
+d
+explain select distinct a1 from t1 where a2 = 'b';
+id	select_type	table	type	possible_keys	key	key_len	ref	rows	Extra
+1	SIMPLE	t1	range	NULL	idx_t1_1	130	NULL	5	Using where; Using index for group-by
+select distinct a1 from t1 where a2 = 'b';
+a1
+a
+b
+c
+d
 drop table t1;
 drop table t2;
 drop table t3;
diff --git a/mysql-test/t/group_min_max.test b/mysql-test/t/group_min_max.test
index 76f959b983b..6731be615fd 100644
--- a/mysql-test/t/group_min_max.test
+++ b/mysql-test/t/group_min_max.test
@@ -482,7 +482,6 @@ select a1,a2,b from t2 where (a2 >= 'b') and (b = 'a') group by a1,a2,b;
 select a1,a2,b,c from t2 where (a2 >= 'b') and (b = 'a') and (c = 'i121') group by a1,a2,b;
 select a1,a2,b from t2 where (a1 > 'a') and (a2 > 'a') and (b = 'c') group by a1,a2,b;
 
-
 --
 -- DISTINCT queries
 --
@@ -526,6 +525,7 @@ select distinct a1,a1 from t1;
 select distinct a2,a1,a2,a1 from t1;
 select distinct t1.a1,t2.a1 from t1,t2;
 
+
 --
 -- DISTINCT queries with GROUP-BY
 --
@@ -641,6 +641,17 @@ explain select a1,a2,count(a2) from t1 group by a1,a2,b;
 explain select a1,a2,count(a2) from t1 where (a1 > 'a') group by a1,a2,b;
 explain select sum(ord(a1)) from t1 where (a1 > 'a') group by a1,a2,b;
 
+#
+# BUG#11044: DISTINCT or GROUP BY queries with equality predicates instead of MIN/MAX.
+#
+
+explain select a1 from t1 where a2 = 'b' group by a1;
+select a1 from t1 where a2 = 'b' group by a1;
+
+explain select distinct a1 from t1 where a2 = 'b';
+select distinct a1 from t1 where a2 = 'b';
+
+
 drop table t1;
 drop table t2;
 drop table t3;
diff --git a/sql/opt_range.cc b/sql/opt_range.cc
index b3301d17655..5732e156a7c 100644
--- a/sql/opt_range.cc
+++ b/sql/opt_range.cc
@@ -7612,8 +7612,8 @@ void cost_group_min_max(TABLE* table, KEY *index_info, uint used_key_parts,
   *records= num_groups;
 
   DBUG_PRINT("info",
-             ("records=%u, keys/block=%u, keys/group=%u, records=%u, blocks=%u",
-              table_records, keys_per_block, keys_per_group, records,
+             ("table rows=%u, keys/block=%u, keys/group=%u, result rows=%u, blocks=%u",
+              table_records, keys_per_block, keys_per_group, *records,
               num_blocks));
   DBUG_VOID_RETURN;
 }
@@ -8120,6 +8120,15 @@ int QUICK_GROUP_MIN_MAX_SELECT::get_next()
       DBUG_ASSERT((have_max && !have_min) ||
                   (have_max && have_min && (max_res == 0)));
     }
+    /*
+      If this is a just a GROUP BY or DISTINCT without MIN or MAX and there
+      are equality predicates for the key parts after the group, find the
+      first sub-group with the extended prefix.
+    */
+    if (!have_min && !have_max && key_infix_len > 0)
+      result= file->index_read(record, group_prefix, real_prefix_len,
+                               HA_READ_KEY_EXACT);
+
     result= have_min ? min_res : have_max ? max_res : result;
   }
   while (result == HA_ERR_KEY_NOT_FOUND && is_last_prefix != 0);
@@ -8146,9 +8155,8 @@ int QUICK_GROUP_MIN_MAX_SELECT::get_next()
     QUICK_GROUP_MIN_MAX_SELECT::next_min()
 
   DESCRIPTION
-    Load the prefix of the next group into group_prefix and find the minimal
-    key within this group such that the key satisfies the query conditions and
-    NULL semantics. The found key is loaded into this->record.
+    Find the minimal key within this group such that the key satisfies the query
+    conditions and NULL semantics. The found key is loaded into this->record.
 
   IMPLEMENTATION
     Depending on the values of min_max_ranges.elements, key_infix_len, and
@@ -8232,9 +8240,7 @@ int QUICK_GROUP_MIN_MAX_SELECT::next_min()
     QUICK_GROUP_MIN_MAX_SELECT::next_max()
 
   DESCRIPTION
-    If there was no previous next_min call to determine the next group prefix,
-    then load the next prefix into group_prefix, then lookup the maximal key of
-    the group, and store it into this->record.
+    Lookup the maximal key of the group, and store it into this->record.
 
   RETURN
     0                    on success